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

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,

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

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