Pythagorean Identity

Definition

The Pythagorean Identity is a fundamental relation in trigonometry that states $\sin^2(\theta) + \cos^2(\theta) = 1$. It holds for any angle $\theta$ and is derived from the Pythagorean Theorem applied to the unit circle.

5 Must Know Facts For Your Next Test

1. $\sin^2(\theta) + \cos^2(\theta) = 1$ is always true, regardless of the value of $\theta$.
2. It can be used to derive other trigonometric identities.
3. The identity is based on a right triangle inscribed in the unit circle.
4. You can express $\sin^2(\theta)$ as $1 - \cos^2(\theta)$ and $\cos^2(\theta)$ as $1 - \sin^2(\theta)$.
5. This identity helps in simplifying complex trigonometric expressions.

Review Questions

• What does the Pythagorean Identity state?
• How can you express $\sin^2(\theta)$ using the Pythagorean Identity?
• Why is the Pythagorean Identity considered fundamental in trigonometry?

Related terms

Unit Circle: A circle with a radius of one centered at the origin of a coordinate plane. It is used to define trigonometric functions.

Sine Function: A function that gives the y-coordinate of a point on the unit circle. Denoted as $\sin(\theta)$.

Cosine Function: A function that gives the x-coordinate of a point on the unit circle. Denoted as $\cos(\theta)$.