Even-odd identities

Definition

Even-odd identities are trigonometric identities that describe the symmetry properties of trigonometric functions. An even function satisfies $f(-x) = f(x)$, while an odd function satisfies $f(-x) = -f(x)$.

5 Must Know Facts For Your Next Test

1. Cosine and secant are even functions: $\cos(-x) = \cos(x)$ and $\sec(-x) = \sec(x)$.
2. Sine, tangent, cotangent, and cosecant are odd functions: $\sin(-x) = -\sin(x)$, $\tan(-x) = -\tan(x)$, $\cot(-x) = -\cot(x)$, and $\csc(-x) = -\csc(x)$.
3. Even-odd identities help simplify trigonometric expressions by reducing negative angles to positive angles.
4. These identities are useful for verifying more complex trigonometric identities.
5. Knowing whether a function is even or odd can assist in graphing the function by identifying its symmetry.

Review Questions

• What is the even-odd identity for sine?
• How does the even-odd identity for cosine differ from that of sine?
• Why are even-odd identities useful in simplifying trigonometric expressions?

Related terms

Pythagorean Identities: Identities derived from the Pythagorean theorem that relate the squares of sine, cosine, and other trigonometric functions.

Sum and Difference Formulas: Formulas that express trigonometric functions of sums or differences of angles in terms of products of trigonometric functions.

Reciprocal Identities: Identities expressing basic relationships between a function and its reciprocal: $\sin x = \frac{1}{\csc x}$, $\cos x = \frac{1}{\sec x}$, etc.