Prove that the sine of a sum can be expressed as a linear combination of sine and cosine

by RoRi

August 1, 2015

Given $C, \alpha \in \mathbb{R}$ prove that there exist $A,B \in \mathbb{R}$ such that

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

Proof. For $C, \alpha \in \mathbb{R}$ we compute,

$\begin{align*} C \sin (x + \alpha) &= C ( \sin x \cos \alpha + \sin \alpha \cos x) \\ &= (C \cos \alpha) \sin x + (C \sin \alpha) \cos x. \\ \end{align*}$

Since sin and cos are defined for all $\alpha \in \mathbb{R}$ , we have $C \cos \alpha, C \sin \alpha \in \mathbb{R}$ . Hence, the requested real numbers are

$A = C \cos \alpha, \qquad B = C \sin \alpha. \qquad \blacksquare$

Stumbling Robot

Prove that the sine of a sum can be expressed as a linear combination of sine and cosine

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