The figure shows a unit circle and a radius with the lengths of its 𝑥- and 𝑦-components. Use the Pythagorean theorem to derive an identity connecting the lengths one, cos 𝜃, and sin 𝜃.
So what I’ve done to help us see a little bit better is blown up the triangle that we’ve got formed. So we can see we’ve got a right-angle triangle. And we know that it’s a right-angle triangle because we’ve got horizontal and a vertical component. And they meet and they are perpendicular to each other. So therefore, I’ve drawn the right angle sign in here.
Then, we’ve got an angle 𝜃. And we’ve got three sides. We’ve got one, sin 𝜃, and cosine 𝜃 or cos 𝜃. So the question tells us to use the Pythagorean theorem. So let’s remind ourselves what that is.
Well, the Pythagorean theorem states that 𝑎 squared plus 𝑏 squared equals 𝑐 squared. And this is where 𝑐 is the hypotenuse, which is the longest side opposite the right angle, and 𝑎 and 𝑏 at the other two sides. To use the Pythagorean theorem, we must have a right angle, which we do. So that’s fine.
So then, what we’re gonna do is we’re gonna label our sides. So we’ve got 𝑐. It’s going to be our hypotenuse, which is opposite the right angle, and then 𝑎 and 𝑏. It doesn’t matter which way round where they are. So I’ve just written 𝑎 as where sin 𝜃 is and 𝑏 as where cos 𝜃 is. So therefore, if we substitute this into the Pythagorean theorem, we’re gonna get sin 𝜃 squared plus cos 𝜃 squared is equal to one squared. And the way we write sin 𝜃 squared is sin squared 𝜃 and similarly with cos squared 𝜃.
So therefore, we can say that the identity connecting the lengths one, cos 𝜃, and sin 𝜃 is sin squared 𝜃 plus cos squared 𝜃 is equal to one.