We want to explain how the identity cos^2 θ+sin^2 θ=1 relates to the Pythagorean Theorem. To do so, let's first consider a unit circle with an angle θ. Recall that the length of the radius is 1 in a unit circle.
The length of the leg on the x-axis is the x-coordinate of P, which is cos θ. The length of the other leg is the y-coordinate of P, which is sin θ.
P(x,y) = P(cos θ, sin θ)
We will now focus on the right triangle. Notice that the length of the hypotenuse is equal to the length of the radius. Therefore, the triangle has a hypotenuse that measures 1.
By the Pythagorean Theorem, the sum of the squares of the lengths of the legs is equal to the length of the hypotenuse squared. Therefore, for our triangle, the sum of the squares of cosθ and sinθ is equal to the square of 1.
Pythagorean Theorem
Given Identity
a^2+ b^2= c^2
( cosθ)^2+( sinθ)^2= 1^2
⇓
cos^2θ+sin^2θ=1
As we can see, we are able to obtain the given identity using the Pythagorean Theorem. Therefore, it is called a Pythagorean Identity.
