Trigonometric identity

Definition

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variable within its domain. These identities are used to simplify expressions and solve trigonometric equations.

5 Must Know Facts For Your Next Test

1. The Pythagorean identities include $\sin^2(x) + \cos^2(x) = 1$, $1 + \tan^2(x) = \sec^2(x)$, and $1 + \cot^2(x) = \csc^2(x)$.
2. The angle sum and difference identities, such as $\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)$, help in calculating the sine or cosine of a sum or difference of angles.
3. Double-angle identities, like $\cos(2x) = \cos^2(x) - \sin^2(x)$, are useful in simplifying expressions involving multiple angles.
4. Trigonometric identities can be proved using algebraic manipulations and known values of trigonometric functions at specific angles.
5. Identities like $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and $\cot(x) = \frac{1}{\tan(x)}$ assist in converting between different trigonometric functions.

Review Questions

• What is the Pythagorean identity involving sine and cosine?
• How would you use the angle sum identity to find $\sin(75^\circ)$?
• What is the double-angle formula for cosine?

Related terms

Pythagorean Identity: An identity that relates the squares of sine and cosine: $\sin^2(x) + \cos^2(x) = 1$.

Angle Sum Identity: $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$ is one example; it helps compute the sine or cosine of sums/differences.

Double-Angle Identity: $\cos(2x)=\cos^2(x)-\sin^2(x)$ simplifies expressions involving double angles.