Products of Trigonometric Functions into Sums with Werner Formulas

Text

Write the following quantities as sums of sines and/or cosines:

(a) $\cos(2x)\cos\left(x-\dfrac\pi3\right)$

(b) $\sin(2x)\cos\left(x-\dfrac\pi3\right)$

(c) $\sin(2x)\sin\left(x-\dfrac\pi3\right)$

Solution

We apply the Werner formulas with $A=2x$ and $B=x-\dfrac\pi3$, so that $A+B=3x-\dfrac\pi3$ and $A-B=x+\dfrac\pi3$.

(a) Using $\cos A\cos B = \tfrac12\left[\cos(A-B)+\cos(A+B)\right]$: $\cos A\cos B = \frac12\left[\cos\!\left(x+\frac\pi3\right)+\cos\!\left(3x-\frac\pi3\right)\right].$

(b) Using $\sin A\cos B = \tfrac12\left[\sin(A+B)+\sin(A-B)\right]$: $\sin A\cos B = \frac12\left[\sin\!\left(3x-\frac\pi3\right)+\sin\!\left(x+\frac\pi3\right)\right].$

(c) Using $\sin A\sin B = \tfrac12\left[\cos(A-B)-\cos(A+B)\right]$: $\sin A\sin B = \frac12\left[\cos\!\left(x+\frac\pi3\right)-\cos\!\left(3x-\frac\pi3\right)\right].$

$\boxed{\text{the three sums above}}$