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Pythagorean Identity: a non-technical explanation

Written by tutor Jeffery D.

Before you learn about trig identities, there are few things you need to know. No, really. As you have probably experienced when studying other maths, you know that math builds on itself. So, please take a moment to review these few concepts to be sure that you understand what they mean. Trust me on this, otherwise you will just have to go back and read it later.

Remember that when reading mathematics, we need to read more slowly and carefully. Do not feel ashamed if you have to read a passage more than once, or even more than five times. Read it out loud, or have someone read it to you if it helps. Math takes concentration and effort. If you utilize those tools you can know whatever math you want to know. So, read slowly and repeat if necessary. Ok. Go!

Pythagorean Theorem

Ok, now that you have those down, we can get started by thinking about the Pythagorean Theorem.

Pythagoras was a guy who lived a long time ago. He and a bunch of his buddies lived together and studied, among a few other things, mathematics. They believed mathematics are divine. Pythagoras had a theorem that has been used over and over again to find side lengths of right triangles. The formula is simple

a2 + b2 = c2

Where a and b are the non-hypotenuse sides, and c is the hypotenuse. There are several proofs for this theorem. Just do a search!

So you may be asking what this has to do with trig identities. Well, what I am going to show you is where the equation comes from

sin(θ)2 + cos(θ)2 = 1

First let me tell you, in very non-technical terms how it is used. Very basically, we use this “identity” when we have some big scary equation++ to solve, derivative to take, or integral to compute. This equality allows us to replace something we can do very little about with something that goes nicely along the path of our calculation. If that does not make sense now, just remember those words. Because when you get to that problem, you can remember them and it will all make sense. Ok? So just keep that in mind for now. We are trying to replace complicated, unworkable things, with more friendly, but equivalent things.

++ Did you know that when you are frightened (perhaps by a big scary equation) it is possible for your left brain (where some of your math skills are stored) to stop working so well. If you get frightened on a test, remember to take some deep breaths and close your eyes. Calm yourself down and start with what you know. Stay calm and breathe deep!

The Unit Circle

If we take a circle (remember I told you we would need this definition and formula?) that is centered at (0, 0), and with a radius of 1, we have the so-called unit circle! We use the unit circle to allow us to represent angles. These angles are can be referred to by their coordinates in the rectangular coordinate system. That is their x and y values. For example, an angle of 0 degrees (0 radians for those using radians) has an x coordinate of 1 and a y coordinate of 0 on the unit circle. The ordered pair (another term for xy coordinates) for an angle of 0 degrees is (1, 0). Another example to look at on the unit circle is an angle of 90 degrees (π/2 radians). This an angle with coordinates (0, 1).

Ok now we are ready to go! From here we can see and believe and then understand several fundamental trigonometric identities.

Now think about this. On the unit circle we call the x and y coordinates cosine and sine. The cosine function is described by the x-coordinate and the sine function is described by the y coordinate.

From the example above, the cosine of an angle with 0 degrees is what? The answer is 1. And that means the sine is what? That’s right (or if you didn’t get guess, read that paragraph again... slower though!) the sine of an angle 0 degrees is 0!

The unit circle is designed, is specifically designed, with a radius of 1 (to learn more about why, look into things that are said to be “unit length” or other mathy things with the word unit in front. You will notice a similarity.) Each pair of x and y (cosine and sine) coordinates are a length of 1 from the origin.

Ok. Get ready because now I am going to show through a series of steps why one thing equals the other thing, therefore (mathy word) this thing will equal to this other thing and hence we can use this to solve that. Are you ready? Ok. Here goes.

We have Pythagoras's Theorem

a2 + b2 = c2

If I let b = sin(θ), and a = cos(θ), and I let c = 1,

**This is a point where you might stop and say... What? Why?

So, remember, we are saying that sine is the y coordinate, and cosine is the x coordinate. On a triangle, the x coordinate would be the horizontal length, and the y coordinate would be the vertical length. That leaves 1 (our radius) as our hypotenuse.

So, remember, we are saying that sine is the y coordinate, and cosine is the x coordinate. On a triangle, the x coordinate would be the horizontal length, and the y coordinate would be the vertical length. That leaves 1 (our radius) as our hypotenuse.

If you need more explanation. Stop reading. Get a piece of paper and writing utensil (or if you are on a nice sunny beach with a warm breeze, just a stick and some sand will do!). Ok. Now, draw a right triangle. I am assuming you know what that is because you followed the instructions above!

Now that you have drawn your triangle, label your hypotenuse with the number 1, and also the letter c.

Next, label your horizontal side with three things. It should say this cosine = x = a.

Finally label your vertical side with three things. It should say this sine = y = b.

Ok. Now continue reading. . .

Then I get

cos(θ)2 + sin(θ)2 = 12

***Ok, here let me just say the symbol θ is a Greek symbol we use in math. It is called “theta” and we usually use it to stand for “some angle”. So, here both of the angles theta must be the same. Otherwise, we would use a different symbol inside sine then we did for cosine. Get it? We can use any variable we want. Like t, x, y, or whatever.

Now back to the circle formula. The unit circle has a formula that is

(x-h)2 + (y-k)2 = r2

With h and k = 0, and r = 1. In other words,

x2 + y2 = 1

Now... look at the triangle you drew!

Here is something that is will take us out of the realm of the unit circle. All we have to do is multiply both sides of the equation by the same thing. Then, we can get a general formula. You do know (if not I am going to tell you) that if I multiply both sides of the equation by the same thing it is the same thing? Well, that is essentially what it is... So, watch what happens.

Let’s multiply both sides by 2, just because 2 is another number that is not 1.

2(x2) + 2(y2) = 2
(2 multiplied by 1 is just 2)

Indeed I can multiply that by any number say, r, since r is for radius.
So

rx2 + ry2 = r

Now I will just substitute my cosine and sine function since we know we can

rcos(θ)2 + rsin(θ)2 = r

When r = 1, we have

cos(θ)2 + sin(θ)2 = 1

*1 squared is just 1 right? That’s why I didn’t say 1 squared and all.

Now do some experimentation. Plug the cosine and sine of angles to see that this formula works! Now we are ready to study the rest of the Pythagorean identities, the quotient identities, and the reciprocal identities using right triangle trigonometry!

Pythagorean Identity Practice Quiz

Is the following equation correct?
sin(30°)2 + cos(90°)2 = 1

A. Yes
B. No
The correct answer here would be B.

True or false? The sine of some angle theta is the y coordinate of that angle on the unit circle.

A. True
B. False
The correct answer here would be A.

What will happen if I divide my formula by 1?

A. The signs change from positive to negative (and from negative to positive).
B. Nothing happens.
The correct answer here would be B.

Is the following equation correct?
2sin(t)2 + 2cos(t)2 = 4

A. Yes
B. No
The correct answer here would be B.

Is the following equation correct?
2sin(x)2 + 2cos(x)2 = 2

A. Yes
B. No
The correct answer here would be A.

How can an angle be characterized on the unit circle?

A. By sine and cosine, its coordinates
B. By the length of the legs of the angle
The correct answer here would be A.
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