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Express a function of sine as a linear combination of sine and cosine

by RoRi

Find A,B \in \mathbb{R} such that

    \[ 3 \sin \left( x + \frac{\pi}{3} \right) = A \sin x + B \cos x \]

for all x \in \mathbb{R}.

Here we just need to compute,

    \begin{align*} 3 \sin \left( x + \frac{\pi}{3} \right) &= 3 \left( \sin x \cos \frac{\pi}{3} + \sin \frac{\pi}{3} \cos x \right) \\ &= 3 \left( \frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x \right) \\ &= \frac{3}{2} \sin x + \frac{3 \sqrt{3}}{2} \cos x. \end{align*}

So, we have,

    \[ A = \frac{3}{2}, \qquad B = \frac{3 \sqrt{3}}{2}. \]

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