Sections 9.3 & 9.4 - Multiple-Angle and Product/Sum Formulas

Section Objectives

1. Use double-angle formulas to rewrite trigonometric expressions.
2. Use power-reducing formulas to rewrite trigonometric expressions.
3. Use product-to-sum and sum-to-product formulas to rewrite trigonometric equations.

Double-angle Formulas

The double-angle formulas follow immediately from the sum formulas.

• $\sin(2u) = 2 \sin u \, \cos u$
• $\cos(2u) = \cos^2 u - \sin^2 u = 2\cos^2 u -1 = 1 - 2 \sin^2 u$
• $\displaystyle \tan(2u) = \frac{2\tan u}{1 - \tan^2 u}$

Examples

• Find all solutions: $\quad 2 \cos x + \sin 2x = 0$

• Find $\sin 2\theta$ if $\cos \theta = 5/13$ and $\theta$ is in the 4th quadrant.

• Determine a formula for $\sin 3x$.

Power-reducing and Half-angle Formulas

These formulas are easily derived from the double-angle formulas.

• $\displaystyle \sin^2 u = \frac{1-\cos 2u}{2}$
• $\displaystyle \cos^2 u = \frac{1+\cos 2u}{2}$
• $\displaystyle \tan^2 u = \frac{1-\cos 2u}{1+\cos 2u}$

Examples

• Rewrite $\sin^4 x$ in terms of 1st powers of sines and/or cosines.

• Derive a half-angle formula for $\sin(x/2)$.

Product-to-sum formulas

These formulas are easy to derive from the sum and difference formulas.

• $\sin u \, \sin v = \frac{1}{2}[\cos(u-v)-\cos(u+v)]$
• $\cos u \, \cos v = \frac{1}{2}[\cos(u-v)+\cos(u+v)]$
• $\sin u \, \cos v = \frac{1}{2}[\sin(u+v)+\sin(u-v)]$
• $\cos u \, \sin v = \frac{1}{2}[\sin(u+v)-\sin(u-v)]$

Examples

• Rewrite $\cos 5x \, \sin 4x$ as a sum or difference.

• Derive a sum-to-product formula and use it to solve $\sin 5x + \sin 3x = 0$.