Mathematics · Ch 3 — Trigonometric Functions

Summary

Angle measurement: Radian measure is standard; $1 \text{ radian} = \frac{180^\circ}{\pi}$ . Arc length $l = r\theta$ , area of sector $= \frac12 r^2\theta$ .
Trigonometric ratios: For an angle $\theta$ in standard position, $\sin\theta = \frac{y}{r}$ , $\cos\theta = \frac{x}{r}$ , $\tan\theta = \frac{y}{x}$ ( $x \neq 0$ ), with reciprocals $\csc\theta$ , $\sec\theta$ , $\cot\theta$ .
Signs in quadrants: All ratios positive in QI; only $\sin$ and $\csc$ in QII; only $\tan$ and $\cot$ in QIII; only $\cos$ and $\sec$ in QIV.
Fundamental identities: $\sin^2\theta + \cos^2\theta = 1$ , $1 + \tan^2\theta = \sec^2\theta$ , $1 + \cot^2\theta = \csc^2\theta$ .
Compound angles: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ ; $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ ; $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ .
Double-angle formulas: $\sin 2A = 2\sin A \cos A$ ; $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ ; $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$ .
Transformation formulas: Sum-to-product: $\sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}$ , $\sin C - \sin D = 2\cos\frac{C+D}{2}\sin\frac{C-D}{2}$ , $\cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}$ , $\cos C - \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2}$ .
Product-to-sum: $2\sin A \cos B = \sin(A+B) + \sin(A-B)$ , $2\cos A \sin B = \sin(A+B) - \sin(A-B)$ , $2\cos A \cos B = \cos(A+B) + \cos(A-B)$ , $2\sin A \sin B = \cos(A-B) - \cos(A+B)$ . …