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Calculus is a complex branch in mathematics that deals with the study of change. Knowledge of how functions change over certain periods or conditions can only be answered with methods derived from calculus. There are 3 core topics which comprise calculus: Limits, Derivation and Integration. Limits describe how functions approach values. Derivation describes the rate of change at certain values of a function. Integration describes the area under the curve of a function. The beauty in calculus lies in how these 3 core topics relate to one another.
\"},\"riemann-sums\":{\"title\":\"Riemann Sums\",\"sections\":[{\"text\":\"...\"}]},\"proof-of-the-fundamental-theorem-of-calculus-part-i\":{\"title\":\"Proof of the Fundamental Theorem of Calculus Part I\",\"sections\":[\"Proving the fundamental theorem of calculus consists of two parts. Interestingly, the 'first part' was proved secondly, and the 'second part' was proved first. However, it makes much more sense for any calculus student to learn the first part's proof, then the second one, despite the order that they were proven. So let's jump right into the first part!
The first part of the fundamental theorem of calculus connects the idea of a derivative and an integral. It states that the derivative of an integral returns the original function. In mathematical notation: \\\\[\\\\frac{d}{dx}\\\\left[\\\\int f(x) \\\\, dx\\\\right] = f(x)\\\\]This can also be used to say that the integral of f(x) (area under the curve) is just the antiderivative of f(x), or in simpler terms, a function g(x) where it's derivative of f(x). As we can see, proving this theorem will give us a gateway into actually solving integrals, one of the hardest tasks in calculus.
But why would we even consider a relationship in the first place? Well, let's consider a very general function, f(x).
Observe f(x) and A(x), where A(x) is the area function. Basically, we are assuming that the area under any curve can be considered some function of x. As we see with the blue line to the right, changing that also changes the overall area of the function. If f(x) is decreasing, then the added area to A(x) will become less and less large. Here, we can intuitively see a connection to the derivative of f(x) and A(x). It just so happens that the area function IS the antiderivative of f(x), which is less intuitive to see.
Lets begin with the proof:
Consider A(x) and f(x) where A(x) is the area function to f(x). We are to prove that A'(x) = f(x).
Let's begin with the definition of a derivative, just to see if we can work with that somehow: \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{A(x+h) - A(x)}{h} \\\\right] \\\\]
Notice A(x + h) and A(x). These appear to be the difference of two areas. Let's represent this graphically.
As we see, A(x + h) - A(x) must be some definite area, but what? Here, we will use interesting logic in our proof. The first thing we notice is that the value of f(x) at point x shown is at a minimum, and f(x+h) is at a maximum for the interval x to x + h. That seems unique, why isn't x + h moved a bit more to the right, where it wouldn't be a maximum of the interval? Well, the key is that the expression A(x + h) - A(x) is a limit, so h is meant to be a slight nudge to x. It seems a lot more like a nudge on the graph to show the idea of the area under a curve. Since h approaches 0, it is meant to be a \u201csufficiently small\u201d value. This means that either f(x) or f(x + h) is a max or a min, depending if the function is increasing or not, because the distance between x and x + h will approach 0. Although it doesn't really matter as we will see, the function is increasing so f(x + h) will be the max. and f(x) will be the min.
Now with that settled, why does that even matter? Well, we know the area between x and x + h is an area, and that f(x) is the lowest point on the graph, and f(x + h) is the highest. Now, we know that the area in the interval is some area value, A. Let's simplify the area for A greatly. Let's say that the area for A is just the area of some rectangle with the width of the interval h. Then, A = h * f(c), where f(c) is some height between f(x) and f(x + h). Ok\u2026 but how does that help us? Well, we know that f(x) is a minimum, so f(x) * h MUST be smaller than the actual area for the interval, because all other points on the graph are greater than f(x). So, if you view the interval as the sum of a bunch of rectangles, we just took the smallest rectangle and spread it across the interval, meaning it must be smaller. With similar logic, since f(x + h) is a maximum, it MUST be larger than the actual area. With that, we know that f(c) must be somewhere between f(x) and f(x + h). Since our function is continuous, we know that every value must exist between f(x) and f(x + h), which means every value must exist between f(x) * h and f(x + h) * h. This means that area A must exist in the form f(c) * h, where c \u2208 [x, x + h]. \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{A(x+h) - A(x)}{h} \\\\right] \\\\]
Back to the actual math, we've essentially just logically proven that A(x + h) - A(x) = f(c) * h. Let's substitute that in! \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ \\\\frac{f(c) * h}{h} \\\\right] \\\\] \\\\[ A'(x) = \\\\lim_{h \\\\to 0} \\\\left[ f(c) \\\\right] \\\\]
Now, we know that c \u2208 [x, x + h] then, \\\\[ x \\\\leq c \\\\leq x + h \\\\]
But, h\u21920, so: \\\\[ x \\\\leq c \\\\leq x \\\\]
Which means c = x, meaning f(c) = f(x), solving the limit.
A'(x) = f(x), which is what we intended to prove.
\"]},\"proof-of-the-fundamental-theorem-of-calculus-part-ii\":{\"title\":\"Proof of the Fundamental Theorem of Calculus Part II\",\"sections\":[\"There is a problem with the first part of our theorem. But\u2026 didn't we just prove it true? We did, but not to where we can do anything meaningful. What do we mean by this?
Note that we just proved A'(x) = f(x), meaning the derivative of the area function of a graph is f(x). But what precisely is the area function? Where does it begin, and where does it conclude? How can it be effectively expressed? The graph used in proving the first part of our fundamental theorem was convenient because it had a definite starting point. However, most graphs do not. Fortunately, we can choose the starting point of a graph without a clear beginning to be anywhere. But what implications does this have for our area function?
To answer this, let's consider the derivative of x2:
\\\\[ \\\\frac{d}{dx}\\\\left[ x^2 \\\\right] = 2x \\\\]Cool! What about x2 + 1?
\\\\[ \\\\frac{d}{dx}\\\\left[ x^2 + 1 \\\\right] = 2x \\\\]x2 + C?
\\\\[ \\\\frac{d}{dx}\\\\left[ x^2 + C \\\\right] = 2x \\\\]As we see, many things can share a derivative. Since the derivative of a constant is always 0, the antiderivative of a function always has \\\"+ C\\\" after it, indicating that the antiderivative of a function is actually a family of functions that all differ from a constant value.
So, if A(x) is just the antiderivative of f(x), which we had just proven, then A(x) = F(x) + C. F(x) is just denoting the antiderivative function for A(x). It actually makes sense why a C needs to be there for such a general area function.
Whether we start our area function at D or E, all the areas to the left of both D and E are the same. This means the area functions for both SHOULD be the same, just differing by some constant, C. This is called the indefinite integral.
\\\\[ \\\\int f(x) \\\\, dx = F(x) + C \\\\]The explanation for the notation will be shown in a different article. Here, we are essentially finding an area function for f(x), but we have no bounds on that function. The changing of bounds will just differ by a constant C, shown above.
We want to know how to evaluate integrals WITH bounds, however. The good thing is, we already know it should really include the area function. Let's say we start our integral at defined point a, and it goes to some variable point x.
\\\\[ A(x) = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]The variable t here is a dummy variable, because we are trying to show that x could be any point on the graph f(t), which originally would be called f(x). The \\\"x\\\"-axis would be called the \\\"t\\\"-axis to make things clearer. Essentially it's just changing the name of variables so as to not get the x in f(x) confused with the x on the integral.
We know that A(x) is the antiderivative of f(x), so A(x) = F(x) + C. Thus:
\\\\[ F(x) + C = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]The important thing to note here is that we defined one boundary for the integral, at A. this means that we should be able to somehow solve for C, as C can no longer be just any constant. This integral with one bound at A should no longer be a family of antiderivatives, but just one specific antiderivative. But how can we solve it? Well, we can do a sneaky trick. Let's make x some defined point B, essentially finding the area of some interval.
Then:
\\\\[ C = \\\\int_{a}^{b} f(t) \\\\, dt - F(b) \\\\]That didn't really help us. But since we start at A, C must be the same, for any other arbitrary endpoint we pick. Then, how can we get rid of the nasty integral? Let's just make b = a! It's sneaky, but x can technically just be a. Now, an integral from a to a is just clearly 0, because there is no area. Thus, C = -F(a). Now, we've essentially discovered the second part of the fundamental theorem of calculus.
\\\\[ \\\\begin{array}{l} F(x) + C = \\\\int_{a}^{x} f(t) \\\\, dt \\\\\\\\ C = -F(a) \\\\end{array} \\\\]Therefore:
\\\\[ F(x) - F(a) = \\\\int_{a}^{x} f(t) \\\\, dt \\\\]We are pretty much done here, but the proof is shown for an interval between defined point A to defined point B, so we can just let x = a. This also allows us to not have to use the dummy variable t anymore and just make that x.
\\\\[ \\\\int_{a}^{b} f(x) \\\\, dx = F(b) - F(a) \\\\]Simple, yet impactful. Basically, it doesn't matter at all what happens between a and b, the value for the integral on an interval is just F(b) - F(a).
\"]}}")}}]); \ No newline at end of file diff --git a/build/static/js/329.94a86816.chunk.js b/build/static/js/329.94a86816.chunk.js deleted file mode 100644 index 2039625..0000000 --- a/build/static/js/329.94a86816.chunk.js +++ /dev/null @@ -1 +0,0 @@ -"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[329],{329:function(e){e.exports=JSON.parse('{"Resources":["algebra","trigonometry","calculus","proofs"]}')}}]); \ No newline at end of file diff --git a/build/static/js/404.fedbda21.chunk.js b/build/static/js/404.fedbda21.chunk.js deleted file mode 100644 index 7d427b6..0000000 --- a/build/static/js/404.fedbda21.chunk.js +++ /dev/null @@ -1 +0,0 @@ -"use strict";(self.webpackChunkmathinfo=self.webpackChunkmathinfo||[]).push([[404],{404:function(s){s.exports=JSON.parse('{"Summary":{"text":"Trigonometry deals with the study of angles. The etymology of the word also provides insight into what branch of math it encompasses, as \\"trigon\\" refers to triangles, or a shape consisting of 3 angles. Although trigonometry is a sub-branch of geometry, it deserves its own in-depth section as trigonometry appears and is necessary to understand for many other aspects of mathematics. Trigonometry will introduce many new ideas with 2 core functions that incorporate angles called sine and cosine that all relate back to the triangle. The inner-workings of trigonometry include right angle trig., general angle trig, trigonometric function analysis, and even complex trigonometry.
"},"right-triangle-trigonometry":{"title":"Right Triangle Trigonometry","sections":["Trigonometry, as in the name, deals with the study of triangles, angles, ratios, trigonometric functions, graphs and a lot more. It is one of the most important topics in Mathematics and is the gateway for numerous other advanced concepts. While math revolves around Trigonometry, Trigonometry revolves around right triangles. A major reason for this is that only right triangles satisfy the Pythagorean Theorem, which states a2 + b2 = c2, where sides a and b are the legs, while side c is the hypotenuse. One of the key aspects of right triangles is their strict vertical and horizontal side due to the 90\xb0 angle. This also means side c is essentially the slope (rise / run) of the triangle. Let\'s take a look at this right triangle:
","With just this diagram, we can identify many different ratios between the sides. There are a total of 6 ratios, each a trigonometric function. These operations are designated as functions because they completely rely on the angle between both sides.
Let\'s focus on angle \u03b1. Starting with the sine function, we know sin = opposite / hypotenuse. Applying that to angle \u03b1, we find that sin(\u03b1) = a / c, because side a is opposite to angle \u03b1, and side c is the hypotenuse of the triangle. Sine works the exact same way with any other angle. For example, sin(\u03b2) = b / c, as side b is opposite to angle \u03b2, and side c is again the hypotenuse of the triangle. Now that we know all about sine, let\'s look at the other core trigonometric functions:
\u03b8 for any given angle:
Sine -> sin(\u03b8) = opposite / hypotenuse
Cosine -> cos(\u03b8) = adjacent / hypotenuse
Tangent -> tan(\u03b8) = opposite / adjacent
Swapping the numerator and denominator of these core functions gives us their reciprocals:
These 6 functions are all extremely important, with relationships to be further elucidated on as we go through trigonometry. For now though, it\'s important to simply understand where these functions come from and how they\'re used in relation to right triangles.
","Trigonometry will help us solve triangles like these. Our goal is to find side length c, but we aren\'t given both sides a and b. So what is side c? Which trigonometric function will encompass both side length11 and side c to return the angle? Looking at the 42\xb0 angle, we can see the side with length 11 is adjacent to it, and side c is the hypotenuse of the triangle. Going back to the trigonometric functions, cosine conveniently uses both the adjacent and hypotenuse sides of a right triangle.
Here cos(42\xb0) is just a constant, as cos(42\xb0) will be the same for all triangles with a 42\xb0 angle. Using a calculator equipped with the cosine function, we arrive at c \u2248 14.8.
"]},"right-angle-trigonometry-test-problem":{"title":"Right Angle Trigonometry Test Problem","sections":["Solution:Let y = the base of the entire triangle.
tan(30\xb0) = 50 / y
y * tan(30\xb0) = 50
y = 50 / tan(30\xb0)
Let z = the base of the inner right triangle, z = y - x.
tan(60\xb0) = 50 / z
z * tan(60\xb0) = 50
z * tan(60\xb0) = 50
x = y - z
x = 50 / tan(30\xb0) - 50 / tan(60\xb0)
x \u2248 57.7
How can you extend the use of trigonometry to non-right angle triangles with trigonometric functions? The key is in breaking up the \u201cGeneral Triangle\u201d into right triangles and seeing what can be done from there. Let\'s look at the general triangle, where every angle, not just two, are unknown.
By dropping an altitude at the height to the base of the general triangle, we have created two right triangles. Looking more closely, we can observe a relationship between the two triangles.
sin(B) = h / aThen, we can solve for the height.
a * sin(B) = hTaking advantage of their equality, we can merge the equations into the following:
a * sin(B) = b * sin(A)And finally, dividing both sides by ab yields us our final result.
sin(B) / b = sin(A) / aRealizing that we placed the altitude at an arbitrary angle, we could have rotated the triangle and placed it anywhere else for the same relationship. Therefore, sin(C) / c must also be equivalent.
\u2234 sin(A) / a = sin(B) / b = sin(C) / cThis very simple proof establishes the theorem known as The Law of Sines.
So what kinds of triangles can we solve with this?
","Solving triangles refers to finding every angle and side length associated with that triangle. AAS - Angle Angle Side, means you know two angles and a side, with the order of appearance being angle-angle-side. Because the direction of rotation is arbitrary to its properties, SAA is functionally the same thing. ASA means you know two angles and a side in between. Looping back to AAS/SAA triangles, they are the easiest to solve.
To find missing angle C in the above description, simply subtract the known angles by 180\xb0 as all triangle degrees add to 180\xb0. C = 110. Finding the other sides is simply a matter of using The Law of Sines.
sin(A) / a = sin(B) / b = sin(C) / c, meaning the angle maintains a ratio with its opposite side, enabling us to find a missing angle/side assuming we have its counterpart and a full angle/side pair.
sin(30\xb0) / a = sin(40\xb0) / 11Solving for angle C would follow the same process.
","SSA Triangles are more tricky to solve, as they lead to ambiguity which could result in two possible ways to create the triangle.
We can set up:
sin(70\xb0) / 10 = sin(B) / 9Here, we must use the inverse sine function. Basically, if sin(\u1340) gives us a side length, then sin-1(side length) gives us our angle. The only thing is, two possible angles satisfy the sine inverse in all cases. When you plug sin-1(9sin(70\xb0) / 10) into the calculator, you will get one value, the PRINCIPLE value. An explanation for all of this will be in later articles regarding trigonometric functions. sin-1(9sin(70\xb0) / 10) \u2248 57.7. Our other possible angle is simply 180 - PRINCIPLE ANGLE, in this case, 180 - 57.7 = 122.3. (For those who understand the unit circle or are revisiting from a later article, we are essentially reaching the equivalent sin(\u1340) value from QII or QIII.)
To see if both of these triangles are possible, let\'s try to get the 3rd value and see if it all makes sense.
180 - 57.7 - 70 = 52.3We cannot have negative angles, so we can reject the 122.3\xb0 triangle. If it yielded a positive third angle, we would have to solve for the rest of our values for BOTH the triangles, and simply have two possibilities.
"]},"general-angle-trigonometry-ii":{"title":"General Angle Trigonometry II","sections":["The Law of Sines is an important relationship in the general triangle, but there\'s another core relationship we can implement when dealing with triangles, the Pythagorean Theorem. Consider the general triangle with an altitude dropped, creating two right triangles composing the general triangle. From side c, let\'s call x the base length of the smaller triangle and c - x the base length of the larger inner triangle.
Instead of relying on sine, let\'s use cosine on angle B to try to do something with that x variable.
cos(B) = x / aNow we have x in terms of our side lengths and angles of our general triangle. As we can see, we have two right triangles, so we can create some relationships using the Pythagorean Theorem. Note that we want to find an end relationship that doesn\'t contain variables such as h or x, as these aren\'t the side lengths or angles of our triangle.
h2 + x2 = a2To remove h, we can set these two equations equal to one another.
a2 - x2 = b2 - (c2 - 2cx + x2)Remember, x = a * cos(B) so to remove x, substitute this in.
a2 + c2 - 2cb * cos(B) = b2Since this is a general triangle, variable names can be changed and swapped as they are arbitrary, the only thing that has to stay is the relationship between the cosine angle and the side.
Therefore, to make it look better, we can change this to: a2 + b2 - 2ab * cos(\u03b3) = c2. The cosine angle stays opposite to side c, with the previous case opposite to side b.
This is called The Law of Cosines, for its use of the cos function instead of the sin function.
If you notice, The Law of Cosines is actually just an extension of the Pythagorean Theorem! When cos is 90\xb0, cos(90\xb0) = 0. This removes the -2ab term and simply gives us a2 + b2 = c2, which is pretty much a special case of The Law of Cosines.
","In this case, The Law of Sines is unusable, leaving us to rely on The Law of Cosines.
We can find the missing side with The Law of Cosines.
92 + 82 - 2(9)(8)cos(40\xb0) = c2We know all three sides, so to find an angle we can simply use The Law of Sines. We do NOT have to worry about the possibility of there being two triangles because if you know the lengths of all three sides, there can be only one possibility for what the angles can be.
","To solve SSS triangles, we can use The Law of Cosines. To eliminate ambiguity when solving, solve for the angle of the largest side first. This removes the possibility of a second triangle because solving from the longest side gives us the largest angle.
52 + 62 - 2(5)(6)cos(x) = 72Proceed with The Law of Sines to get the other angles.
"]},"the-unit-circle-i":{"title":"The Unit Circle I","sections":["To truly understand trigonometry, one must comprehend how angles work, understanding the unit circle, its relation to angles, and a novel angle measurement called radians.
Angles reach a maximum of 360\xb0, completing a full rotation in circular motion. This range from 0 - 360\xb0 exists because 360 is a conveniently divisible number (by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.). While practical for everyday use, it lacks the mathematical precision we desire.
Consider a circle with a radius of 1, making the circumference C = 2\u03c0. This equates a full 360-degree rotation to 2\u03c0 radians, a unit used in advanced trigonometry. To convert between radians and degrees, multiply by 180/\u03c0 or \u03c0/180, respectively.
Now, observe the unit circle on the coordinate plane with an angle traversing it:
Two angles are measured, originating from the right x-axis and extending outward. The blue and red lines, termed terminal rays, gauge the angle size. Notably, the red terminal ray measures externally, while the blue one measures internally, following the counterclockwise motion. For example, angle a is around 135\xb0 or 3\u03c0/4 radians, while angle b is greater. For angles exceeding 360\xb0, they continue counterclockwise, such as 500\xb0 being equivalent to 140\xb0 after a full rotation. Negative angles rotate clockwise; -1\xb0 equals 359\xb0. This insight allows us to divide the unit circle into four quadrants.
Every \u03c0/4 (90\xb0) defines a new quadrant, with x and y lines acting as quadrantal separators.
Now that we understand this improved angle system and its application around the unit circle, let\'s explore its significance and connection to right triangles. Unlike general right triangles, special triangles like the 45-45-90 and 30-60-90 offer exact angle and side measurements.
These special triangles, created using the Pythagorean theorem and specific base splits, provide exact values without trigonometry. Placing these triangles within the unit circle, with a hypotenuse of length 1, allows us to compress them by dividing by 2 and 3, respectively. This results in three triangles per quadrant: 45-45-90, 30-60-90, and 60-30-90. This process enables the mapping of coordinates for various angles on the unit circle, forming a helpful diagram for memorization.
How does this relate to trigonometric functions? The crucial point lies in the fact that the hypotenuse of these triangles is always 1. Thus, sin(\u03b8) = opp, where \\"opp\\" is the y-coordinate, and cos(\u03b8) = x-coordinate. This insight leads to the realization that coordinates on the circle are simply (cos(\u03b8), sin(\u03b8)). For instance, with our special triangle values, sin(\u03c0/4) = sqrt(2)/2.
Examining the unit circle reveals essential insights into trigonometric functions:
Sine and cosine values range between -1 and 1.
Sin and cosine values are equal at two values: \u03c0/4, 5\u03c0/4
In QI and QII, sin is positive; in QIV and QI, cos is positive.
As sin increases, cos decreases, and vice versa. Two different angles return the same value, either in the top or bottom half for sin and left or right for cos (sin(\u03c0 - \u1340) = sin\u1340, cos(2\u03c0 - \u1340) = cos\u1340).
Finally, recognizing that cos(\u1340) = x and sin(\u1340) = y, tan(\u1340), the slope, is found as y/x, making tan(\u1340) = sin(\u1340)/cos(\u1340). Discovering these relationships is termed finding identities, a concept explored further in our trigonometry studies.
Let\'s explore some intriguing relationships between trigonometric functions using the unit circle while maintaining clarity and coherence. First, let\'s examine the coordinates of points on circles with radii greater or less than 1.
sin(x) = y * r
r * sin(x) = y
r * cos(x) = x
Simply multiply by r, the radius. The ratios are still the same.
(r * sin(x), r * cos(x))Consider the unit circle.
Here, four angles are presented with colored terminal rays. The red and blue rays are reflected across the y-axis, and the blue and yellow across the x-axis. If we examine cos(blue) and cos(yellow), they are the same. Yellow is simply the negative angle of blue; thus, cos(-x) = cos(x). With sine, sin(-x) = -sin(x). Examining the relationship between blue and red, or yellow and purple, we observe that sin(\u03c0 - x) = sin(x), and cos(\u03c0 - x) = -cos(x).
We also observe that cos(2\u03c0 - x) = cos(x), and sin(2\u03c0 - x) = -sin(x). Knowing that tan(x) is the slope, when cos or sin is negative, tan will be negative. Therefore, tan(x + \u03c0) = tan(x), tan(x - \u03c0) = -tan(x), and tan(-x) = -tan(x). These relationships are intuitive and can be quickly deduced from the unit circle.
Now, let\'s explore the relationship between sine and cosine.
When rotating 90\xb0, cos(x), which is x, would be rotated completely vertically upward, becoming sine, and sine would become cosine. Starting from the triangle in the first quadrant, upon rotation, the cosine and sine values swap places, and the sine becomes negative. This process is repeated in the second triangle, yielding the following relationships:
sin(x + \u03c0/2) = cos(x)If we examine the tangent values, the cosine and sines are swapped, and one value becomes negative. Adding a right angle, \u03c0/2, yields the negative reciprocal of the original. Therefore:
tan(x + \u03c0/2) = tan(x - \u03c0/2)More succinctly:
tan(x + \u03c0/2) = -cot(x)These relationships showcase the beauty of trigonometric functions.
"]},"the-unit-circle-iii":{"title":"The Unit Circle III","sections":["While we are acquainted with the representations of cos and sin on the unit circle, it\'s essential to explore how all trigonometric functions manifest and derive identities from these visual connections.
Acknowledging that cos and sin correspond to our x and y values, the question arises: how do we express tan as a length measure? Leveraging the unit circle\'s triangle with a hypotenuse of 1, we can establish a right angle from the (cos, sin) point.
Upon closer inspection, we observe that tan represents that line, given that tan is opp/adj, and adj is 1. This elucidates the name \\"tan,\\" signifying its tangent relationship to the circle. Additionally, it visually explains why tan(\u03c0/2) and tan(\u03c0/4) are undefined, extending infinitely without intersecting the x-axis. To determine sec, representing adj/opp, we leverage the triangle formed with tan, realizing that sec must denote the hypotenuse of that triangle.
Applying a similar process, we can ascertain cotangent.
Similarly, the derivation of cosecant follows a comparable procedure.
Exploring the relationships between trigonometric functions through the Pythagorean theorem, we unveil crucial connections:
sin2(x) + cos2(x) = 1
tan2(x) + 1 = sec2(x)
cot2(x) + 1 = csc2(x)
sin2(x) + (sec(x) - cos(x))2 = tan2(x)
sec2(x) + csc2(x) = (tan(x) + cot(x))2
These relationships are commonly known as the Pythagorean Identities. Among them, the most significant is sin2(x) + cos2(x) = 1, a fundamental identity that merits in-depth study.
"]},"using-the-pythagorean-identity-sine-and-cosine":{"title":"Using the Pythagorean Identity: Sine and Cosine","sections":["Let\'s review the previously derived Pythagorean Identity: sin2(x) + cos2(x) = 1.
Given: sin(\u03b8) = 2/3, \u03b8 in QII
22 + x2 = 32
4 + x2 = 9
x2 = 5
x = \xb1sqrt(5)
Cosine must be negative in QII, therefore we reject positive sqrt(5) / 3.
x = -sqrt(5)While this represents one method of solving, leveraging the Pythagorean Identity provides a more streamlined approach.
sin2(\u03b8) + cos2(\u03b8) = 1Reject positive sqrt(5) / 3.
cos(\u03b8) = -sqrt(5) / 3
Then we can get the adjacent side from how we calculate cosine:
cos(\u03b8) = adj/hypWhile we are familiar with the six trigonometric functions through the unit circle, let\'s explore what these functions look like when graphed. Our x value will represent the angle value in radians, and the y value will be the corresponding return value.
Let\'s examine the special values in the sin function that are rational: 0 - 0, \u03c0/2 - 1, \u03c0 - 0, 3\u03c0/2 - -1, and 2\u03c0 - 0. We observe this cycling motion every 2\u03c0 radians, known as the period. Let\'s visualize this graphically.
Upon graphing and examination, a beautiful cycling pattern emerges.
Since sin(x) = cos(x - \u03c0/2), we anticipate a \u03c0/2 shift to the right for the cosine graph.
","Now, let\'s apply standard function transformations to see how these functions change with applications such as f(x) = A sin[B(x - h)] + k. This applies to both sin and cos.
Let\'s examine each transformation individually:
When looking at sin and cos, which cycle between 1 and -1, we notice a change by a value from 1 in opposite directions from the average line, 0. Therefore, |A| is the amplitude, representing the distance the maximum and minimum values deviate from the midline point.
Note: A negative sign on A could flip everything across the x-axis. A = (max - min) / 2, as this finds the average distance from the midline.
B represents the period. In a normal wave, it repeats every 2\u03c0. When B increases, the entire thing compresses, resulting in a period of 2\u03c0 / B.
h denotes the horizontal shift, and k is the vertical shift. To find k, you can find the actual average for the midline: (max + min) / 2.
","EXAMPLES:
f(x) = 3cos(2x) + 1
g(x) = 2sin(\u03c0x + \u03c0) - 3
For f(x):
k is 1, making the midline 1, and A is 3, so it will extend up and down by 3. The 2x means this will happen during the period \u03c0.
For g(x):
A is 2. No reflection is needed. B is \u03c0, so the period is 2\u03c0 / \u03c0, which is just 2. x is shifted left by 1, and the midline is -3.
What problems can we model with our knowledge of the graphs of sines and cosines?
Consider the word problem: A Ferris wheel with a diameter of 520 feet has a low point of 30 feet. Every 30 minutes, it completes 1 rotation (starting at the bottom). Let f(x) model the height of a point on the Ferris wheel.
We begin at the lowest point, and since a cos graph always begins at 1, a negative cos must start at the bottom.
If the diameter is 520 feet, then we go 260 feet up and down from the midline, meaning a = 260.
If a full rotation is 30 minutes, then 30 = 2\u03c0 / B. B = \u03c0 / 15.
To start at 30, instead of the current -260, we need to add 290.
So, f(x) = -260cos(\u03c0 / 15 * x) + 290.
"]},"trigonometric-functions-graphs-and-problems-ii":{"title":"Trigonometric Functions Graphs and Problems II","sections":["We graphed cos and sin, but how would we graph their reciprocal functions, csc and sec? Remember that csc(x) = 1/sin(x) and sec(x) = 1/cos(x). This is very important. Realizing that sin and cos are in the denominator, we can also see that whenever they return zero, the fraction would be 1/0, which is undefined. These occur every \u03c0 period and are known as asymptotes.
Here for the csc graph, these are all the points we know are undefined. What happens when we approach those points? They must either approach infinity or negative infinity, as when the denominator gets smaller, it returns a larger and larger value. When the sin graph is in the positive half of the coordinate plane, it must blow up to infinity, and when the sin graph is negative, it will expand to negative infinity. Let\'s use a sine graph to visualize this:
As we see, some parts stay between 0 and 1 and others go from 0 to -1. Our csc graph will touch the sin graph when it peaks at 1 or -1, as 1/1 = 1, but will diverge from there. In the negative parts, to negative infinity. With this information, we can create our cosecant graph in orange.
The secant graph is done the exact same way. This process helps you graph any sec or csc graph, as you simply graph the sin or cos graph and then apply the asymptote concept.
tan(x) and cot(x) graphs:
tan(x) is undefined at \u03c0/2 + 2\u03c0k and 3\u03c0/2 + 2\u03c0k. Graphing tan is quite simple for this reason. We know when tan(x) goes negative and goes positive, so we can make a graph easily.
With cot(x), we notice that the 0\'s will become the asymptotes, BUT the asymptotes in tan(x) will become the 0\'s for cot(x). These ideas are common to see with reciprocal functions.
Comparing the two graphs, they almost look reflected. This comes from the identity that we previously derived that states tan(x + \u03c0/2) = -cot(x). Also, take note that for these graphs, the period is not 2\u03c0, just \u03c0.
y = 3cot(3\u03c0/2(x)).
Here we would have a stretch of 3, and a period of 3/2. Given that cot(x) has asymptotes at 0, \u03c0, 2\u03c0, etc.., we can solve for the asymptotes by taking both of these base asymptotes and then applying the proper transformations. :
3\u03c0/2(x) = 2\u03c0We can see here that the steps in the asymptotes are in increments of 2/3, something we could also have calculated by knowing that the function repeats every \u03c0. Now, you could also calculate the 0\'s by directly applying transformations, but a much simpler way is to realize that the asymptotes of any trigonometric function are equidistant from the x intercept on the base function. This way, we can see that the 0\'s would be 1/3, 1, etc\u2026 Since this is a cot function, the function will continuously decrease. Given all of these baby steps calculated, try graphing this for yourself, since the challenging parts are out of the way."]},"the-inverse-trigonometric-functions":{"title":"The Inverse Trigonometric Functions","sections":["
The inverse trig functions, instead of returning ratios, give the angle that provides that ratio. The naming of these is quite easy, and simply just adds an \'arc\' to the functions. They can also be named with the inverse sign, what we are familiar with from the right and general angle trigonometry.
arccos(x) OR cos-1(x)
arcsin(x) OR sin-1(x)
arctan(x) OR tan-1(x)
arccsc(x) OR csc-1(x)
arcsec(x) OR sec-1(x)
arccot(x) OR cot-1(x)
For example, arcsin(1/2) = \u03c0/6
Why not also 5\u03c0/6? To make it a function, we have to restrict its domain to the right half of the unit circle, [-\u03c0/2, \u03c0/2]. arctan(x) is also only between [-\u03c0/2, \u03c0/2]. cos however must be defined as the top half of the unit circle,[0, \u03c0]. Similar to the square root, the domain restriction only exists when these functions are used alone, however in trigonometric equations, every angle will be allowed.
","Consider sin(arccos(x)). How would we solve a composite function like this?
sin(arccos(x)) = \u221a(1 - x2)
We can resemble x/1 as a triangle since these are the lengths of the hypotenuse and adjacent.
sin is opp / hyp. So, what is opp? x2 + opp2 = 1. opp = \u221a(1 - x2)
Therefore, sin(arccos(x)) = \u221a(1 - x2)
We can also notice the same process would work for: cos(arcsin(x)).
\u2234 sin(arccos(x)) = \u221a(1 - x2)
cos(arcsin(x)) = \u221a(1 - x2)
What about arccos(sin(x))?
sin(x) = cos(\u03c0/2 - x)
arccos(cos(\u03c0/2 - x))
The arccos and cos would cancel out, leaving \u03c0/2 - x.
\u2234 arccos(sin(x)) = \u03c0/2 - x
arcsin(cos(x)) = \u03c0/2 - x
With trigonometric equations, domain restrictions no longer apply; we consider every possible angle unless specified otherwise. For instance, 3\u03c0 / 4 + 2\u03c0k indicates solutions at 3\u03c0 / 4 and all angles obtained by adding or subtracting multiples of 2\u03c0, where k belongs to the set of integers (k \u220a \u2124).
","2cos(\u1340) = sqrt(3)
cos(\u1340) = sqrt(3) / 2
\u1340 = \u03c0 / 6, but can also be 11\u03c0 / 6 because of the unit circle. You can also go fully 2\u03c0 around the circle, allowing the following (in all these cases k \u220a \u2124).
\u1340 = \u03c0 / 6 + 2\u03c0k, 11\u03c0 / 6 + 2\u03c0k
Let\'s look at something more challenging:
5cos(3\u1340 + 1) = 5Let\'s substitute u for 3\u1340 + 1 to make this simpler.
cos(u) = 1sin(x) = -0.29
arcsin(-0.29) = x
-0.294 \u2248 x, convert this to positive radians by subtracting it from 2\u03c0.
x \u2248 5.989
So how do we find the second angle? Let\'s look at the unit circle.
Blue is the 5.989 radians. To get to its equivalent red angle, we can observe adding 0.294 to \u03c0 to get there.
x \u2248 3.436Let\'s combine the ideas with this equation.
7cos(3\u1340) = 4 on [0, 2\u03c0]
cos(3\u1340) = 4 / 7
u = 3\u1340
cos(u) = 4 / 7
u = arccos(4 / 7) *calc
u = 0.963
u = 2\u03c0 - 0.963
u = 5.320
3\u1340 = 5.320 + 2\u03c0k, 3\u1340 = .963 + 2\u03c0k
\u1340 = 1.773 + 2\u03c0k / 3, 0.321 + 2\u03c0k / 3
Since this is on [0, 2\u03c0], we can account for 3 revolutions, as a single revolution is 2\u03c0k / 3.
\u1340 = 0.961, 7.246, 13.529, 5.320, 11.603, 17.887Let\'s solve our final introductory equation.
65 = -13cos(\u03c0 / 12 * (\u1340 - 5)) + 61Let\'s review the identities we\'ve established, focusing on our Pythagorean identities:
sin2(x) + cos2(x) = 1
tan2(x) + 1 = sec2(x)
1 + cot2(x) = csc2(x)
These become useful when solving trigonometric equations through factoring. Now, let\'s explore a simple example.
2sin2(x) + sin(x) = 0
sin(x)(2sin(x) + 1) = 0
This gives two possible solutions:
sin(x) = 0
sin(x) = -\xbd
Which leads to:
x = \u03c0k, 11\u03c0/6 + 2\u03c0k, 7\u03c0/6 + 2\u03c0k.
","These problems share similarities with quadratics. Now, let\'s tackle a slightly more challenging equation:
2sin2(x) - cos(x) = 1.
Here, we leverage the identity sin2(x) + cos2(x) = 1 by expressing sin2(x) in terms of cos(x):
sin2(x) = 1 - cos2(x)
Substitute this into the original equation:
2(1 - cos2(x)) - cos(x) = 1
Simplify to get:
-2cos2(x) - cos(x) = -1
Rearrange and factor like a quadratic:
-2cos2(x) - cos(x) + 1 = 0
2cos2(x) + cos(x) - 1 = 0
Factor further:
(2cos(x) - 1)(cos(x) + 1) = 0
This gives two solutions:
cos(x) = 1/2
cos(x) = -1
Therefore, x can be expressed as:
x = \u03c0/3 + 2\u03c0k, 5\u03c0/3 + 2\u03c0k, \u03c0 + 2\u03c0k
"]},"fundamental-trigonometric-identities-i":{"title":"Fundamental Trigonometric Identities I","sections":["Let\'s consider the unit circle with two points, P and Q. Q is represented by angle \u03b2, and P by \u03b1.
Here, Q = (cos(\u03b2), sin(\u03b2)), and P = (cos(\u03b1), sin(\u03b1)).
Now, P\' = (cos(\u03b1 - \u03b2), sin(\u03b1 - \u03b2)), and Q\' = (1, 0).
A crucial observation is that, since we rotated the triangle, the line PQ is equivalent to P\'Q\'.
We can set up the distances using the distance formula:
PQ = sqrt((cos(\u03b1) - cos(\u03b2))2 + (sin(\u03b1) - sin(\u03b2))2)Setting them equal to each other:
(cos(\u03b1) - cos(\u03b2))2 + (sin(\u03b1) - sin(\u03b2))2 = (cos(\u03b1 - \u03b2) - 1)2 + sin2(\u03b1 - \u03b2)Simplifying the equation:
2 - 2cos(\u03b1)cos(\u03b2) - 2sin(\u03b1)sin(\u03b2) = (cos(\u03b1 - \u03b2) - 1)2 + sin2(\u03b1 - \u03b2)This derivation yields the formula: cos(\u03b1 - \u03b2) = cos(\u03b1)cos(\u03b2) + sin(\u03b1)sin(\u03b2), a valuable identity in sum and difference identities.
Now, let\'s explore other identities by setting \u03b2 = -\u03b3:
cos(\u03b1 + \u03b3) = cos(\u03b1)cos(\u03b3) - sin(\u03b1)sin(\u03b3)Thus, we obtain:
cos(\u03b1 + \u03b2) = cos(\u03b1)cos(\u03b2) - sin(\u03b1)sin(\u03b2)These are the cosine sum and difference identities.
For sine, we can use the substitution trick for sin(x) = cos(x - \u03c0/2) and -sin(x) = cos(x + \u03c0/2).
Let \u03b1 = \u03c0/2 - \u03b3:
sin(\u03b3 + \u03b2) = sin(\u03b3)cos(\u03b2) + cos(\u03b3)sin(\u03b2)Thus, we derive:
cos(\u03b1 + \u03b2) = cos(\u03b1)cos(\u03b2) - sin(\u03b1)sin(\u03b2)These are our sum and difference identities, crucial for solving various trigonometric equations.
"]},"fundamental-trigonometric-identities-ii":{"title":"Fundamental Trigonometric Identities II","sections":["We previously discovered these identities:
cos(\u03b1 + \u03b2) = cos(\u03b1)cos(\u03b2) - sin(\u03b1)sin(\u03b2)
cos(\u03b1 - \u03b2) = cos(\u03b1)cos(\u03b2) + sin(\u03b1)sin(\u03b2)
sin(\u03b1 + \u03b2) = sin(\u03b1)cos(\u03b2) - cos(\u03b1)sin(\u03b2)
sin(\u03b1 - \u03b2) = sin(\u03b1)cos(\u03b2) - cos(\u03b1)sin(\u03b2)
How can we use them to discover MORE identities? Well, it\'s quite simple! Notice that you can cancel out certain terms by adding equations:
cos(\u03b1 + \u03b2) = cos(\u03b1)cos(\u03b2) - sin(\u03b1)sin(\u03b2)
cos(\u03b1 - \u03b2) = cos(\u03b1)cos(\u03b2) + sin(\u03b1)sin(\u03b2)
cos(\u03b1 + \u03b2) + cos(\u03b1 - \u03b2) = 2cos(\u03b1)cos(\u03b2)
(cos(\u03b1 + \u03b2) + cos(\u03b1 - \u03b2)) / 2 = cos(\u03b1)cos(\u03b2).
We can repeat this process and we can find identities for all pairs of multiplying sine and cosine, which gives us:
cos(\u03b1)cos(\u03b2) = (cos(\u03b1 + \u03b2) + cos(\u03b1 - \u03b2)) / 2
sin(\u03b1)sin(\u03b2) = (cos(\u03b1 - \u03b2) - cos(\u03b1 + \u03b2)) / 2
sin(\u03b1)cos(\u03b2) = (sin(\u03b1 + \u03b2) + sin(\u03b1 - \u03b2)) / 2
These are our products to sum identities.
The product to sum identities are particularly useful anytime we want to convert multiplication of trigonometric functions into addition and subtraction. This will become useful when we eventually put these new identities to use. Now that we know product -> sum identities, how would we go upon getting sum -> product identities? The key is in rearranging \u03b1 and \u03b2 in our product to sum identities.
Given: sin(\u03b1)cos(\u03b2) = (sin(\u03b1 + \u03b2) + sin(\u03b1 - \u03b2) / 2
let u = \u03b1 + \u03b2, v = \u03b1 - \u03b2. We must get u and v in terms of \u03b1 and \u03b2 so we can change the entire equation to be in terms of u and v.
v = \u03b1 - \u03b2
u = \u03b1 + \u03b2
Adding these,
v + u = 2\u03b1
(u+v) / 2 = \u03b1
Subtracting instead of adding,
v - u = -2\u03b2
(u - v) / 2 = \u03b2
Substituting into our given, we get:
sin((u+v) / 2)cos((u - v) / 2)) = (sin(u) + sin(v)) / 2
2sin((u+v) / 2 )cos((u - v) / 2)) = sin(u) + sin(v)
If we let w = -v, we can find the subtraction variant.
2sin((u - w) / 2)cos((u + w) / 2)) = sin(u) - sin(w)
Using the exact same process, we can find these identities for sine and cosine, which are as follows:
2cos((u + v) / 2))cos((u - v) / 2)) = cos(u) + cos(v)
2sin((u + v) / 2))sin((u - v) / 2)) = cos(u) - cos(v)
\u2234
2sin((u + v) / 2 )cos((u - v) / 2 )) = sin(u) + sin(v)
2sin((u - w) / 2 )cos((u + w) / 2 )) = sin(u) - sin(w)
2cos((u + v) / 2))cos((u - v) / 2)) = cos(u) + cos(v)
2sin((u + v) / 2))sin((u - v) / 2)) = cos(u) - cos(v)
The sum and difference identities, the most fundamental of the more complex trigonometric identities, can be used in many ways, including finding exact values to previously unknown angles.
Let\'s consider the sum identity for sin(x).
sin(\u03b1 + \u03b2) = sin(\u03b1) * cos(\u03b2) + sin(\u03b2) * cos(\u03b1)We can find unknown angle values by adding or subtracting known ones. For example, let\'s consider \u03c0/6 and \u03c0/4. Summing these together, we get 5\u03c0/12, an angle that we currently have no clue what the exact value is.
sin(5\u03c0/12) = sin(\u03c0/4) * cos(\u03c0/6) + sin(\u03c0/6) * cos(5\u03c0/12)What about a base fraction, like \u03c0/12? Knowing the exact value of that could help us find the exact value of a lot more angles. Here, we can use the difference formula.
What subtracts to equal \u03c0/12? \u03c0/3 and \u03c0/4 do!
sin(\u03b1 - \u03b2) = sin(\u03b1) * cos(\u03b2) - sin(\u03b2) * cos(\u03b1)Certainly isn\'t as nice looking as something like sin(\u03c0/3), which is why these values usually do not need to be memorized. This all goes to show that any trigonometric function will yield an exact value if the angle is in terms of \u03c0, due to the transcendentality of the trigonometric functions and the transcendentality of \u03c0.
sin(5 radians) will never be in exact value (it would be transcendental), but sin(3\u03c0/19) does have an exact value.
","Oftentimes, we find that the sum and difference identities are a bit too general for what we need to deal with. These can simplify to what are known as the double angle identities.
sin(\u03b1 + \u03b2) = sin(\u03b1) * cos(\u03b2) + sin(\u03b2) * cos(\u03b1)What if \u03b1 = \u03b2?
sin(2\u03b1) = sin(\u03b1) * cos(\u03b1) + sin(\u03b1) * cos(\u03b1)Also note that cos(\u03b1 + \u03b2) = cos(\u03b1) * cos(\u03b2) - sin(\u03b1) * sin(\u03b2).
cos(2\u03b1) = cos(\u03b1) * cos(\u03b1) - sin(\u03b2) * sin(\u03b2)These can make it easier to expand out something like sin(3\u03b1).
sin(3\u03b1) = sin(2\u03b1 + \u03b1)Notice that there may be a relationship between the overall degree of the relationship and the inner coefficient of sin(3\u03b1). We will dive deeper into this in later articles.
"]},"using-identities-single-sine-expression":{"title":"Using Identities: Single Sine Expression","sections":["As we journey through mathematics, we may find it very useful to rewrite trigonometric expressions as sine functions. This becomes helpful when solving trigonometric equations in some cases, but also with calculus and graphing. Let\'s see what we can do with a general sine expression.
The general form is A * sin(Bx + C) where A, B and C are constants. Let\'s try to separate this out. We want to work backwards from the answer because we are trying to convert complex expressions to this general form.
Using the sine addition identity, we can expand to A * [sin(Bx) * cos(C) + cos(Bx) * sin(C)]. Notice that sin(C) and cos(C) are constants. Carrying on, let\'s distribute out constant A and move the constants in front.
A * cos(C)sin(Bx) + A * sin(C)cos(Bx)Since A * cos(C) and A * sin(B) are constant values, let\'s rewrite them as M and N respectively.
M * sin(Bx) + N * cos(Bx)Hold on a second. We can use one of our identities to find a relationship between M, N, and A to remove those nasty sine and cosine functions.
M2 + N2 = (A * cos(C))2 + (A * sin(C))2We have two more trivial relationships to uncover:
M = A * cos(C)This means M / A = cos(C) and N / A = sin(C). Let\'s see what we managed to discover here.
M * sin(Bx) + N * cos(Bx) = A * sin(Bx + C)We can now turn any expression that involves adding cosine and sine with the same angle into a single sine function to be solved or graphed!
Let\'s look at the most simple of these expressions: cos(x) + sin(x).
M = 1A = sqrt(2)
B = 1, since x * 1 = x in cos(x) and sin(x)
Here, we see that \u03c0/4 satisfies both of these relationships, so cos(x) + sin(x) = sqrt(2) * sin(x + \u03c0/4).
Let\'s try another!
4sqrt(3) * sin(5x) - 4cos(5x)The value that satisfies both equations is 11\u03c0/6. Thus, 4sqrt(3) * sin(5x) - 4cos(5x) = 8sin(5x + 11\u03c0/6).
"]},"solving-trigonometric-equations-ii":{"title":"Solving Trigonometric Equations II","sections":["Now that we have derived some identities and found some tricks within trigonometry, let\'s solve some equations that we could not previously solve before. Let\'s start with a problem involving sine and cosine.
2sin(4x) - 8cos(4x) = sin(2x) * cos(2x) + 3This looks challenging, but it\'s quite simple with our knowledge of identities. First, we can use the single sine trick, then the product to sum identity. We could also multiply both sides of this equation by 2, and use the double angle identity. Both work here, but generally try to find the fastest solution for when things get harder.
4sin(4x) - 16cos(4x) = 2sin(2x) * cos(2x) + 6Now we can turn this into a single sine.
32 + 162 = A2C = 0.185 rad, where both equations are true.
sqrt(265) * sin(4x + 0.185) = 6Remember, treat everything within sin() as u.
sin(u) = 6 / sqrt(265)u = 0.377 + 2\u03c0n, where n is an integer.
u = 2.764 + 2\u03c0n
Substitute in for u.
Let\'s solve another, simpler equation for some exact values.
sin(x) * sin(2x) + cos(x) * cos(2x) = sqrt(3) / 2Here, use the difference formula for cosine. When solving for equations, we should condense without losing a possible root.
cos(x - 2x) = sqrt(3) / 2Remember that cosine isn\'t affected by that negative value, you can memorize this for efficiency, but it never hurts to sketch the unit circle to see this simple relationship.
cos(x) = sqrt(3) / 2Here x = \u03c0/6, 11\u03c0/6.
Let\'s solve one more.
We can\'t use previous methods in this article, but we luck out because there isn\'t a constant term. This means if we can get a product form and factor, we can solve it. Good thing there\'s an identity for that!
2sin((x + 3x) / 2) * cos((x - 3x) / 2) = cos(x)x = \u03c0/12, 5\u03c0/12 (add \u03c0 to navigate through the interval)
x = 13\u03c0/12, 17\u03c0/12
In the end, x = \u03c0/2, 3\u03c0/2, \u03c0/12, 5\u03c0/12, 13\u03c0/12, 17\u03c0/12.
We\'ve moved beyond fundamental trigonometric identities, which are essential for most trigonometry topics, to more advanced identities useful in calculus. The \\"Power Reduction\\" identity simplifies trigonometric functions raised to high powers, making them easier to solve or graph, using the Pythagorean identity:
sin2(\u03b8) + cos2(\u03b8) = 1For higher-degree functions, we use the double angle identity for cosine:
cos(2\u03b8) = cos2(\u03b8) - sin2(\u03b8)From the Pythagorean identity, we know:
cos2(\u03b8) = 1 - sin2(\u03b8)Substituting, we get:
cos(2\u03b8) = 1 - 2sin2(\u03b8)Solving for sin2(\u03b8):
sin2(\u03b8) = (1 - cos(2\u03b8)) / 2Similarly, solving for cos2(\u03b8):
cos2(\u03b8) = (1 + cos(2\u03b8)) / 2Using these, we derive the half-angle identities. Let \ud835\udf03 = \u03b1 / 2 to substitute and simplify:
cos2(\u03b1 / 2) = (1 + cos(\u03b1)) / 2Taking square roots:
cos(\u03b1 / 2) = sqrt((1 + cos(\u03b1)) / 2)Thus, we have:
sin2(\u03b8) = (1 - cos(2\u03b8)) / 2Algebra involves arithemitic, variables, and various rules under its name. However, at its heart, Algebra is the study of equality. Equations, which encompass the study of any relationship, are at the centerpiece of Algebra\'s vast domain. Topics include the study of linear relationships, quadratics, polynomials, rational expressions, radicals, and ways to simplify. Many consider it to be mathematics most important branch of study, as Algebra is one of the few branches that cross its roots with every other topic in math. Its expansiveness allows it be taught from the middle school level to graduate school lectures.
"},"the-basis-of-quadratics":{"title":"The Basis of Quadratics","sections":["In mathematics, specifically algebra, solving equations for certain variables has been at the forefront in the exploration of new ideas for thousands of years. Quadratic equations have been known to mathematicians since ancient times, however it is believed that a formula to solve all quadratic equations was discovered much later by Arab mathematicians in 9th Century A.D. But what exactly is a quadratic equation?
To be considered quadratic, the highest degree term in a polynomial must be x2. Some examples of this are: x2, 9x2 + 6x + 7, y = 3x2 - 9, etc. Each of these are considered quadratic expressions / equations. But, how do we solve them?
First, there is solving linear equations: which is something we may be more familiar with. Linear equations are always in the form: y = mx + b. Solving these usually consists of using simple operations to isolate the variable x. Let\'s walk through an example.
9x + 15 = 6
9x = -11 (subtract 15 from both sides)
x = -11/9 (divide 9 from both sides)
Quite simple! Moving on, it is standard for linear and quadratic equations like these to first be made equal to 0 for simplicity.
ax + b = 0 {a \u2260 0} (if a was 0, there would be no x in the equation!)
How do we solve this? There are no numbers for coefficients! Before we panic, we\'ll simply use the same process we did before to solve this general case.
ax + b = 0
ax = -b (subtract b from both sides)
x = -b/a (divide by a on both sides)
We have just created an equation that can be used to solve any linear equation when equal to 0! Try it yourself with the first example, just make sure the equation equals 0.
While this is great and all, we are still trying to discover how to solve quadratic equations, not linear ones. When looking at quadratics, we can see that they either come in 1, 2 or 3 terms.
1 term: x2 = 0
2 terms: x2 + 9 = 0, 3x2 + 5x = 0, etc.
3 terms: 3x2 + 5x + 7 = 0, 11x2 + 6x + 9 = 15 (remember, you can subtract 15 from both sides), etc.
Let\'s attempt to solve all three kinds of these quadratics. One term quadratics are the easiest because they always are 0!
11x2 = 0
x2 = 0 (divide by 11 on both sides)
x = 0 (square root both sides)
Something slightly more challenging would be the two term quadratics, however they are very similar to the first.
5x2 + 7 = 11
5x2 = 4 (subtract 7 from both sides)
x2 = 4/5 (divide by 5 on both sides)
x = +2/\u221a5, -2/\u221a5 (take the square root on both sides, remember this gives us a positive and negative answer)
5x2 + 7x = 0
x(5x + 7) = 0 (factor out an x from the left)
Here we can use the 0 product law, where we know that anything times 0 is 0! So, if we have two separate terms, x and (5x+7) multiplying to equal 0, we know that two cases can arise, where the first term x is 0 OR the second term (5x+7) is 0. This can give us two solutions by breaking up the equation.
x = 0For many, all of this is simply review, however understanding these simpler cases is integral to understanding the most general case: the three term quadratic. ALL quadratics can be expressed like this: ax2 + bx + c = 0 {a \u2260 0}. To get the one and two term quadratics, b and c can both be 0. So how do we solve the generic 3 term?
2x2 + 11x - 2 = 0We can try to move things around and factor things out, but nothing seems to help us. Try things out for yourself! You\'ll see that you won\'t be able to isolate x without a different method of factoring.
To solve this equation, we need to condense it where we have two terms to easily solve, but how do we do that? Let\'s look at the expression (x + y)2 and multiply it out. (x + y)(x + y) using FOIL, gives us x2 + 2yx + y2. An expression like (x + 2)2 becomes x2 + 4x + 4, a three term quadratic that can be factored and condensed! So, if we get any quadratic into x2 + 2yx + y2 form, we might be able to solve quadratics in all cases.
2x2 + 11x - 2 = 0Let\'s rearrange this into something that could be put in x2 + 2yx + y2 form.
x2 + (11/2)x = 1 (added 2 on both sides, then divided everything by 2).
Looking at the coefficient (11/4), it can be expressed in 2y form: 2 * (11/4) where y = 11/4.
x2 + (2)(11/4)x = 1We are getting close to the form we want, we just need to take y2 and add it to this equation. We know y = 11/4, so we should square this and add it to both sides!
2 + (2)(11/4)x + (11/4)2 = 1 + (11/4)2Now, we can factor the left side. Since it\'s in x2 + 2yx + y2 form, let\'s turn it into (x + y)2.
(x + 11/4)2 = 1 + (11/4)2And those are our answers. If we look at the process, we brought the constant term to the other side, divided by the x2 coefficient, and then added a specific constant term to both sides of the equation that allowed us to factor. Considering ax2 + bx + c = 0, it was half of b and then squared: (b/2)2 or b2/4. Adding that then allows us to factor to (x + b/2)2.
This process is called "completing the square" and can be used in all cases. Here is a visualization of the process "completing" the square in a sense.
The large square and the two rectangular areas represent x2 + bx when all added together. With only that, the square isn\'t finished. However, adding the small square area of (b/2)2 completes it into the entire square of the area (x + b/2)2. An example of completeing the square:
3x2 + 6x - 8 = 0 (Step 1: Get x2 + bx alone.)
3x2 + 6x = 8
x2 + 2x = 8/3 (Step 2: Add b2/4 to both sides.)
x2 + 2x + 1 = 11/3 (Step 3: Factor to (x + b/2)2 and solve.)
(x + 1)2 = 11/3
x + 1 = +/- \u221a(11/3)
x = -1 +/- \u221a(11/3)
Perhaps there is a formula to solve all quadratics like linear equations. Now that we know the process, let\'s see if we can solve the classic ax2 + bx + c = 0.
ax2 + bx + c = 0
ax2 + bx = -c
x2 + (b/a)x = -c/a (The b/a coefficient will act like the "b" in our method.)
x2 + (b/a)x + b2/4a2 = -c/a + b2/4a2
[x + (b/2a)]2 = -c/a + b2/4a2 (To make things neater on the right, turn it into a fraction by multiplying -c/a * 4a in the numerator and denominator to get the common base 4a2.)
[x + (b/2a)]2 = (b2 - 4ac)/4a2
x + (b/2a) = \u221a(b2 - 4ac)/\u221a4a2
x + (b/2a) = +/- \u221a[(b2 - 4ac)/2a]
x = -b/2a +/- \u221a[(b2 - 4ac)/2a] (Don\'t forget the common base: 2a.)
x = [-b +/- \u221a(b2 - 4ac)]/2a
We\'ve finally arrived at the formula to solve any quadratic we would like! It\'s called the quadratic formula and it solves for x in ANY quadratic.
4x2 + 9x - 8 = 0 (a = 4, b = 9 and c = -8.)
2x2 - 14 = 0 (a = 2, b = 0 and c = -14.)
x2 = 0 (a = 1, b = 0 and c = 0.)
The quadratic formula is often overkill for many problems, as a lot can be solved with simpler methods such as factoring. However, the quadratic formula is the only one that can solve ALL quadratics.
\\\\[ x = \\\\frac{{-b \\\\pm \\\\sqrt{{b^2 - 4ac}}}}{{2a}} \\\\]"]},"studying-the-quadratic-formula":{"title":"Studying the Quadratic Formula","sections":["In the quadratic formula, the solution equations for the quadratic polynomials of ax2 + bx + c = 0, a != 0, is expressed as x = [-b \xb1 \u221a(b2 - 4ac)] / 2a. At its core, there are the most important parts: \xb1 and b2 - 4ac. The \xb1 gives us the ability for 2 possible solutions, however what lies in the radical: b2 - 4ac, determines the nature of these roots. This is why we refer to it as the "discriminant". When b2 - 4ac = 0, there is always 1 solution to the quadratic, as you either add or subtract 0 in the equation. This is considered the "double root", and is graphically represented with a parabola that touches the x axis only once. When b2 - 4ac > 0, there are always 2 solutions, and when b2 - 4ac < 0, there are 2 complex solutions intuitively. These complex solutions always come in conjugates, as the complex component is either added or subtracted from the real component -b. For example, if given 3 - 6i as a root, you immediately know that 3 + 6i must be the other root.
","It immediately can be noticed that finding the sums or products of roots from polynomial quadratics must always yield real results, as conjugates when added or multiplied must always have their complex components disappear.
(a + bi) + (a - bi) = 2a
(a + bi)(a - bi) = a2 - abi + abi - b2i2
(a + bi)(a - bi) = a2 + b2
What does this mean about the sums and products of all quadratic roots?
Let \u03b1 = the first root, \u03b2 = the second root.
\u03b1 = [-b + \u221a(b2 - 4ac)] / 2a
\u03b2 = [-b - \u221a(b2 - 4ac)] / 2a
\u03b1 + \u03b2 = [-b + \u221a(b2 - 4ac)] / 2a + [-b - \u221a(b2 - 4ac)] / 2a
\u03b1 + \u03b2 = -2b / 2a (discriminants cancel out)
\u03b1 + \u03b2 = -b / a
\u03b1\u03b2 = [-b + \u221a(b2 - 4ac)] / 2a * [-b - \u221a(b2 - 4ac)] / 2a
\u03b1\u03b2 = [-b + \u221a(b2 - 4ac)][-b - \u221a(b2 - 4ac)]/4a2
\u03b1\u03b2 = [b2 - (b2 - 4ac)] / 4a2
\u03b1\u03b2 = 4ac / 4a2
\u03b1\u03b2 = c / a
\u2234 \u03b1 + \u03b2 = -b / a, \u03b1\u03b2 = c / a
It is quite fascinating to see that by adding and multiplying the roots of a quadratic, very simple equations arise from them. These formulas can be used to build certain quadratic equations with certain roots and properties by using the sum and product of the roots. Perhaps the biggest takeaway could be the relationship shown between the coefficient terms and constant terms of a quadratic equation and how exactly they relate to the problem at hand.
Proofs provide the backbone of mathematics, as they rigorously explain why things are the way they are. While much of math has real life application which betters our lives, proofs usually consist of pure logic and mathematical ideas to solely explain and prove some mathematical topic. Given proofs uphold logical soundness, it is considered to be the most complex and difficult branch of mathematics. Proofs serve to better provide a deeper understanding to mathematical concepts as the idea goes against simply providing a formula or statement without evidence for why that it is true.
"},"pythagorean-theorem-and-distance-formula-derivation":{"title":"The Pythagorean Theorem and The Distance Formula Derivation","sections":["The Pythagorean Theorem is one of the most fundamental theorems in all of Mathematics, but little are familiar with the proof of such a formula. This theorem is used to find the length of any diagonal line which gives us a formula for how to calculate the distance between two points in space. But how was it proven?
Let\'s throw away any knowledge of the Pythagorean Theorem and simply try to answer this question: What is the length of this line on the coordinate plane?
Looking at the image, we can see that A = (-6, -3), and B = (7, 6). We can easily calculate distance when the points are on the same dimension, but how do we calculate distance when two dimensions are involved? Looking at A and B, we can see they are 13 units away horizontally and 12 units away vertically. Let\'s represent this as a triangle.
","Here, c is our unknown, b is 13 and a is 12. This creates a right triangle. Right triangles are special for many reasons - a big factor being their properties of similarity.
Above shows how each component angle of each triangle must be all equivalent when Triangle 3 is bisected to where it internally creates Triangle 1 and Triangle 2. What similarity tells us is that the ratios of the side lengths must be equivalent. This is because while the triangles can have different sizes overall, their interior angles dictate the side length proportions. Therefore, when two angles are shown to be equal, the third must also be equal, creating these proportions. For example, in the previous model: a / b = e / d. Right triangles are the only triangles where you can have this and its two components similar, due to the creation of two equal degree angles when bisecting the hypotenuse. You can mess around with these ratios and find many relationships, so let\'s try to expand and discover something we didn\'t expect.
","Consider two ratios: a / c = e / a and b / c = f / b. These ratios are particularly interesting because they contain a and b side lengths of the main triangle twice in their respective equations. By cross multiplying, a2 = ce and b2 = cf. Since c is in both of these equations, let\'s add them together to simplify things: a2 + b2 = ce + cf. It is clear that we can factor out c: a2 + b2 = c(e + f). Looking at the diagram: e + f = c, meaning we can substitute e + f out for c and get a2 + b2 = c2. This relationship is not obvious at all, but can beautifully be derived from very simple similarity relationships. Some good news here is that since c is isolated, we can solve for the diagonal side length of a triangle given only vertical and horizontal lengths, being c = sqrt(a2 + b2). We can finally get that equation for the distance of any two points on a plane.
let D be the distance
C = sqrt(a2 + b2) (Substitute out hypotenuse length c.)
D = sqrt(a2 + b2) (a is our vertical length and b is our horizontal length.)
a = | y1 - y2 |
b = | x1 - x2 |
D = sqrt((y2 - y1)2 + (x2 - x1)2) (The absolute values are removed because the whole thing is squared, and since negative * negative = positive, the order of variables doesn\'t matter.)
Now we can find the distance between A = (-6, -3) and B = (7, 6). With proper substitution we get:
Deriving the Pythagorean Theorem in three dimensions is surprisingly simple, as it builds on the 2D Pythagorean Theorem. The Pythagorean Theorem in two dimensions states: a2 + b2 = c2, where a and b are the legs of a right triangle, and c is the hypotenuse.
From this three dimensional illustration, we want to find what s is from non-angled lines: x, y and z.With help from the Pythagorean Theorem, we can deduce: c2 = x2 + y2. We can also notice that on the rectangular prism, z stays completely vertical. With that information, it is time to use the Pythagorean Theorem on the triangle containing s.
z2 + (sqrt(x2 + y2))2 = s2
x2 + y2 + z2 = s2 (Absolute value signs are not needed when canceling out due to x and y both squared.)
With changing the variable names to match the Pythagorean Theorem, we get: a2 + b2 + c2 = d2. There is a clear pattern when increasing one dimension, and it\'s by simply adding a new variable. To find a distance formula in 3D space, the process is exactly the same:
When going up a dimension, all you need to do is add the subtracted coordinates squared. You can see this in action when going down dimensions. If D = sqrt((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2), decreasing to a single dimension gives us D = sqrt(x2 - x1)2), which when canceled is: D = | x2 - x1 |, the formula for finding the distance between two points on the number line.
Therefore, for n dimensions, the Pythagorean Theorem theoretically would be: a12 + \u2026 + an2 = c2, where n is the amount of dimensions, each term is the distance in its dimension, and c is the hypotenuse.
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' +\n 'Use PropTypes.checkPropTypes() to call them. ' +\n 'Read more at http://fb.me/use-check-prop-types'\n );\n err.name = 'Invariant Violation';\n throw err;\n };\n shim.isRequired = shim;\n function getShim() {\n return shim;\n };\n // Important!\n // Keep this list in sync with production version in `./factoryWithTypeCheckers.js`.\n var ReactPropTypes = {\n array: shim,\n bigint: shim,\n bool: shim,\n func: shim,\n number: shim,\n object: shim,\n string: shim,\n symbol: shim,\n\n any: shim,\n arrayOf: getShim,\n element: shim,\n elementType: shim,\n instanceOf: getShim,\n node: shim,\n objectOf: getShim,\n oneOf: getShim,\n oneOfType: getShim,\n shape: getShim,\n exact: getShim,\n\n checkPropTypes: emptyFunctionWithReset,\n resetWarningCache: emptyFunction\n };\n\n ReactPropTypes.PropTypes = ReactPropTypes;\n\n return ReactPropTypes;\n};\n","/**\n * Copyright (c) 2013-present, Facebook, Inc.\n *\n * This source code is licensed under the MIT license found in the\n * LICENSE file in the root directory of this source tree.\n */\n\nif (process.env.NODE_ENV !== 'production') {\n var ReactIs = require('react-is');\n\n // By explicitly using `prop-types` you are opting into new development behavior.\n // http://fb.me/prop-types-in-prod\n var throwOnDirectAccess = true;\n module.exports = require('./factoryWithTypeCheckers')(ReactIs.isElement, throwOnDirectAccess);\n} else {\n // By explicitly using `prop-types` you are opting into new production behavior.\n // http://fb.me/prop-types-in-prod\n module.exports = require('./factoryWithThrowingShims')();\n}\n","/**\n * Copyright (c) 2013-present, Facebook, Inc.\n *\n * This source code is licensed under the MIT license found in the\n * LICENSE file in the root directory of this source tree.\n */\n\n'use strict';\n\nvar ReactPropTypesSecret = 'SECRET_DO_NOT_PASS_THIS_OR_YOU_WILL_BE_FIRED';\n\nmodule.exports = ReactPropTypesSecret;\n","/**\n * @license React\n * react-dom.production.min.js\n *\n * Copyright (c) Facebook, Inc. and its affiliates.\n *\n * This source code is licensed under the MIT license found in the\n * LICENSE file in the root directory of this source tree.\n */\n/*\n Modernizr 3.0.0pre (Custom Build) | MIT\n*/\n'use strict';var aa=require(\"react\"),ca=require(\"scheduler\");function p(a){for(var b=\"https://reactjs.org/docs/error-decoder.html?invariant=\"+a,c=1;ca||125d?(a.sortIndex=c,f(t,a),null===h(r)&&a===h(t)&&(B?(E(L),L=-1):B=!0,K(H,c-d))):(a.sortIndex=e,f(r,a),A||z||(A=!0,I(J)));return a};\nexports.unstable_shouldYield=M;exports.unstable_wrapCallback=function(a){var b=y;return function(){var c=y;y=b;try{return a.apply(this,arguments)}finally{y=c}}};\n","'use strict';\n\nif (process.env.NODE_ENV === 'production') {\n module.exports = require('./cjs/scheduler.production.min.js');\n} else {\n module.exports = require('./cjs/scheduler.development.js');\n}\n","var map = {\n\t\"./algebra.json\": [\n\t\t573,\n\t\t573\n\t],\n\t\"./calculus.json\": [\n\t\t156,\n\t\t156\n\t],\n\t\"./proofs.json\": [\n\t\t775,\n\t\t775\n\t],\n\t\"./trigonometry.json\": [\n\t\t404,\n\t\t404\n\t]\n};\nfunction webpackAsyncContext(req) {\n\tif(!__webpack_require__.o(map, req)) {\n\t\treturn Promise.resolve().then(function() {\n\t\t\tvar e = new Error(\"Cannot find module '\" + req + \"'\");\n\t\t\te.code = 'MODULE_NOT_FOUND';\n\t\t\tthrow e;\n\t\t});\n\t}\n\n\tvar ids = map[req], id = ids[0];\n\treturn __webpack_require__.e(ids[1]).then(function() {\n\t\treturn __webpack_require__.t(id, 3 | 16);\n\t});\n}\nwebpackAsyncContext.keys = function() { return Object.keys(map); 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It does not describe the direction of the\n * navigation, only that the current index changed.\n *\n * Note: This is the default action for newly created history objects.\n */\n Pop = \"POP\",\n\n /**\n * A PUSH indicates a new entry being added to the history stack, such as when\n * a link is clicked and a new page loads. When this happens, all subsequent\n * entries in the stack are lost.\n */\n Push = \"PUSH\",\n\n /**\n * A REPLACE indicates the entry at the current index in the history stack\n * being replaced by a new one.\n */\n Replace = \"REPLACE\",\n}\n\n/**\n * The pathname, search, and hash values of a URL.\n */\nexport interface Path {\n /**\n * A URL pathname, beginning with a /.\n */\n pathname: string;\n\n /**\n * A URL search string, beginning with a ?.\n */\n search: string;\n\n /**\n * A URL fragment identifier, beginning with a #.\n */\n hash: string;\n}\n\n/**\n * An entry in a history stack. A location contains information about the\n * URL path, as well as possibly some arbitrary state and a key.\n */\nexport interface Location extends Path {\n /**\n * A value of arbitrary data associated with this location.\n */\n state: any;\n\n /**\n * A unique string associated with this location. May be used to safely store\n * and retrieve data in some other storage API, like `localStorage`.\n *\n * Note: This value is always \"default\" on the initial location.\n */\n key: string;\n}\n\n/**\n * A change to the current location.\n */\nexport interface Update {\n /**\n * The action that triggered the change.\n */\n action: Action;\n\n /**\n * The new location.\n */\n location: Location;\n\n /**\n * The delta between this location and the former location in the history stack\n */\n delta: number | null;\n}\n\n/**\n * A function that receives notifications about location changes.\n */\nexport interface Listener {\n (update: Update): void;\n}\n\n/**\n * Describes a location that is the destination of some navigation, either via\n * `history.push` or `history.replace`. May be either a URL or the pieces of a\n * URL path.\n */\nexport type To = string | Partial