Pythagorean Identity

The Pythagorean Identity relates the radius of length one on the unit circle to the squares of the sine and cosine function.
Figure 1: Pythagorean Identity

The Pythagorean Identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is part of the collection of Trigonometric Identities and can be derived using pythagorean’s theorem and the properties of the unit circle[1]. Geometrically, the components of the equation can be visualized by the right triangle below.

Pythagorean Identity

The pythagorean identity can be derived using the trigonometric functions cosine and sine and the right triangle with a hypotenuse of one. This is triangle is best demonstrated with the unit circle, which visualizes all right triangles of hypotenuse one.

Unit Circle Notation

Using the definitions of sine and cosine, we can substitute the variables which represent the adjacent, opposite, and hypotenuse sides of the right triangle into the equations. In this case, the adjacent side corresponds

When we substitute x in for the adjacent side, y for the opposite side, and 1 for the hypotenuse, we can see the adjacent side corresponds directly with the cosine function and the opposite side corresponds with the sine function. Finally, to finish deriving the pythagorean identity, we can combine these facts with Pythagorean’s Theorem, which relates the squares of the sides together.

Pythagorean Theorem

First, convert the theorem into the cartesian coordinate system. This should feel very familiar as it is the equation of a circle.

Then, substitute the corresponding functions above into the equation to get Pythagorean’s identity.


Derive Pythagorean Identity

To derive the pythagorean identity the definition of pythagorean's theorem is combined with the notion of a right triangle placed on the unit circle.

Derive Pythagorean Identity (Unit Circle)

To geometrically derive the pythagorean identity, divide the right triangle into two similar triangles by drawing a line from the right angle corner of the right-triangle perpendicular to its hypotenuse. Then solve for each triangle's length on the hypotenuse.



  1. Derive Pythagorean Identity (Unit Circle)
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