Pythagorean identities

Definition

Pythagorean identities are fundamental trigonometric identities derived from the Pythagorean theorem. They relate the squares of the sine and cosine functions to 1, providing a basis for simplifying and verifying trigonometric expressions.

5 Must Know Facts For Your Next Test

1. The basic Pythagorean identity is $\sin^2(\theta) + \cos^2(\theta) = 1$.
2. Two other important forms are $1 + \tan^2(\theta) = \sec^2(\theta)$ and $1 + \cot^2(\theta) = \csc^2(\theta)$.
3. These identities can be used to derive other trigonometric identities by manipulating their algebraic forms.
4. They are often employed in simplifying complex trigonometric expressions and solving trigonometric equations.
5. Understanding these identities is crucial for verifying more complicated trigonometric identities.

Review Questions

• What is the basic Pythagorean identity involving sine and cosine?
• How can you express secant squared using tangent?
• In what ways can Pythagorean identities simplify solving trigonometric equations?

Related terms

Sine: $\sin(\theta)$ represents the ratio of the opposite side to the hypotenuse in a right triangle.

Cosine: $\cos(\theta)$ represents the ratio of the adjacent side to the hypotenuse in a right triangle.

Tangent: $\tan(\theta)$ represents the ratio of sine to cosine, or opposite side to adjacent side in a right triangle.