# Video: Defining the Pythagorean Identities

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 1, cos π, and sin π.

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### 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 Iβve done to help us see a little bit better is blown up the triangle that weβve got formed. So we can see weβve got a right-angle triangle. And we know that itβs a right-angle triangle because weβve got horizontal and a vertical component. And they meet and they are perpendicular to each other. So therefore, Iβve drawn the right angle sign in here.

Then, weβve got an angle π. And weβve got three sides. Weβve got one, sin π, and cosine π or cos π. So the question tells us to use the Pythagorean theorem. So letβs 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 thatβs fine.

So then, what weβre gonna do is weβre gonna label our sides. So weβve got π. Itβs going to be our hypotenuse, which is opposite the right angle, and then π and π. It doesnβt matter which way round where they are. So Iβve 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, weβre 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.