Express the sine of a sum as a linear combination of sine and cosine

by RoRi

August 1, 2015

Determine constants $C, \alpha$ with $C > 0$ such that

$C \sin (x+\alpha) = -2 \sin x - 2 \cos x$

for all $x \in \mathbb{R}$ .

From the previous exercise, we have the formulas,

$A \sin x + B \cos x = C \sin (x+\alpha)$

for

$\quad C = \sqrt{A^2+B^2}, \quad \text{and }\quad \sin \alpha = \frac{A}{(A^2+B^2)^{1/2}},\quad \cos \alpha = \frac{B}{(A^2+B^2)^{1/2}}.$

Since here we have $A = B = -2$ we compute,

$C = ((-2)^2+(-2)^2)^{1/2} = 2 \sqrt{2},$

and,

$\sin \alpha =\cos \alpha = -\frac{\sqrt{2}}{2} \quad \implies \quad \alpha = \frac{5\pi}{4}.$

Stumbling Robot