Section 9.2 - Sum and Difference Formulas

Section Objectives

1. Use sum and difference formulas to evaluate trigonometric functions.
2. Use sum and difference formulas to verify trigonometric identities.
3. Use sum and difference formulas to solve trigonometric equations.

Sum and Difference Formulas

There are six common sum and difference formulas. Here they are written in compact form:

• $\sin(u \pm v) = \sin u \, \cos v \pm \cos u \, \sin v$
• $\cos(u \pm v) = \cos u \, \cos v \mp \sin u \, \sin v$
• $\displaystyle \tan(u \pm v) = \frac{\tan u \pm \tan v}{1 \mp \tan u \, \tan v}$

Examples for Objective 1

• Find the exact value of $\sin( \pi/12)$.

• Find the exact value of $\tan 75^{\circ}$.

• Write $\sin(\tan^{-1} 1 + \cos^{-1} x)$ as an algebraic expression.

Examples for Objective 2

• Verify that $\displaystyle \sin \! \left( x - \frac{\pi}{2} \right) = -\cos x$.

• Use a sum formula to rewrite $\tan(\theta + 3\pi)$.

• Use a sum formula to expand $\sin(x+h)$.

Examples for Objective 3

• Find all solutions in the interval $[0,2\pi)$: $\quad \sin(x+\pi)-\sin x + 1 =0$

• Find all solutions in the interval $[0,2\pi)$: $\quad \cos(x+\pi/4)-\cos(x-\pi/4)=1$