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I'm having trouble seeing the equality used in Wikipedia's article on simple harmonic motion.

$c_1 \cos(\omega t) + c_2 \sin(\omega t) = A\cos(\omega t- \varphi)$

where $A = \sqrt{c_1^2 + c_2^2}$ and $\tan \varphi = c_2/c_1$, $\omega$ is a constant, and $t$ is the variable in question.

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1 Answer 1

up vote 3 down vote accepted

You have to write

$$\cos(\omega t-\phi)=\cos(\omega t)\cos(\phi)+\sin(\omega t)\sin(\phi)$$

then, equating the right hand side and left hand side, you get

$$c_1=A\cos\phi \qquad c_2=A\sin\phi$$

and it is easy to see that

$$A=\sqrt{c_1^2+c_2^2} \qquad \tan\phi=\frac{c_2}{c_1}.$$

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