AP Precalculus: Trigonometric Identities & Properties
1. Complementary Angle Identities
- \( \sin(90^\circ - x) = \cos x \), \( \cos(90^\circ - x) = \sin x \)
- \( \tan(90^\circ - x) = \cot x \), \( \cot(90^\circ - x) = \tan x \)
- \( \sec(90^\circ - x) = \csc x \), \( \csc(90^\circ - x) = \sec x \)
2. Symmetry & Periodicity
- Even Functions: \( \cos(-x) = \cos x \), \( \sec(-x) = \sec x \)
- Odd Functions: \( \sin(-x) = -\sin x \), \( \tan(-x) = -\tan x \), etc.
- Periodicity: \( \sin(x + 2\pi) = \sin x \), \( \cos(x+2\pi) = \cos x \), \( \tan(x+\pi) = \tan x \)
3. Pythagorean & Reciprocal Identities
- Pythagorean: \( \sin^2 x + \cos^2 x = 1 \)
- \( 1 + \tan^2 x = \sec^2 x \)
- \( 1 + \cot^2 x = \csc^2 x \)
- Reciprocal:
- \( \sin x = \frac{1}{\csc x} \), \( \cos x = \frac{1}{\sec x} \)
- \( \tan x = \frac{1}{\cot x} \), \( \cot x = \frac{1}{\tan x} \)
4. Express a Trig Ratio in Terms of Another
- Use pythagorean and reciprocal identities to go from one ratio to another (e.g., \( \tan x = \frac{\sin x}{\cos x} \), \( \sec x = \frac{1}{\cos x} \))
- Often: you solve for one unknown by substituting known values
5. Sum & Difference Identities
- \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
- \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
- \( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)