### Video Transcript

The figure shows a unit circle and a radius with the lengths of its π₯- and π¦-components. Use the Pythagorean theorem to derive an identity connecting the lengths one, cos π, and sin π.

So what Iβve done to help us see a little bit better is blown up the triangle that weβve got formed. So we can see weβve got a right-angle triangle. And we know that itβs a right-angle triangle because weβve got horizontal and a vertical component. And they meet and they are perpendicular to each other. So therefore, Iβve drawn the right angle sign in here.

Then, weβve got an angle π. And weβve got three sides. Weβve got one, sin π, and cosine π or cos π. So the question tells us to use the Pythagorean theorem. So letβs remind ourselves what that is.

Well, the Pythagorean theorem states that π squared plus π squared equals π squared. And this is where π is the hypotenuse, which is the longest side opposite the right angle, and π and π at the other two sides. To use the Pythagorean theorem, we must have a right angle, which we do. So thatβs fine.

So then, what weβre gonna do is weβre gonna label our sides. So weβve got π. Itβs going to be our hypotenuse, which is opposite the right angle, and then π and π. It doesnβt matter which way round where they are. So Iβve just written π as where sin π is and π as where cos π is. So therefore, if we substitute this into the Pythagorean theorem, weβre gonna get sin π squared plus cos π squared is equal to one squared. And the way we write sin π squared is sin squared π and similarly with cos squared π.

So therefore, we can say that the identity connecting the lengths one, cos π, and sin π is sin squared π plus cos squared π is equal to one.