CEMC Fermat, 2015

Algebra Difficulty 2.4 Junior Find the answer Canada

If $\cos 60^\circ = \cos 45^\circ \cos\theta$ with $0^\circ \leq \theta \leq 90^\circ$ , then $\theta$ equals

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Solution

Since $\cos60^\circ = \frac{1}{2}$ and $\cos 45^\circ = \frac{1}{\sqrt{2}}$ , then the given equation $\cos 60^\circ = \cos 45^\circ \cos \theta$ becomes $\frac{1}{2} = \frac{1}{\sqrt{2}}\cos \theta$ .

Therefore, $\cos \theta = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$ .

Since $0^\circ \leq \theta \leq 90^\circ$ , then $\theta = 45^\circ$ .

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