# How can I express this answer with 3 oscillations in terms of 2 oscillations?

I want to express the equation $\frac{2}{9}\sin(9t) + \frac{9}{56}\cos(5t) + \frac{271}{56}\cos(9t)$ in terms of two oscillations of cosine so it'll have the form:

_cos(_t - _) + _cos(_t)

Is this possible?

Thank you!

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## 1 Answer

Using $$\cos\left(y - x\right) = \sin x \sin y + \cos x \cos y$$ with $y=9t$, $$\frac{2}{9} \sin\left(9t\right)+\frac{271}{56}\cos\left(9t\right) = \cos\left(9t - x\right)$$ where $$x = \tan^{-1} \left(\tan x\right) = \tan^{-1}\left(\frac{\sin x}{\cos x}\right) = \tan^{-1}\left(\frac{2 / 9}{ 271 / 56}\right) = \tan^{-1} \left( \frac{112}{2439}\right).$$

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