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Question Number 38881 by Rio Mike last updated on 30/Jun/18

$$solvefor0⩽\theta ⩽\pi ,theequations$$$${a)}_{}\cos (2\theta -\frac{\pi }{2})=\frac{1}{2}$$$$b)\cos \theta -\sqrt{3}sin\theta =0$$

Answered by MrW3 last updated on 30/Jun/18

$$0⩽\theta ⩽\pi$$$$-\frac{\pi }{2}⩽2\theta -\frac{\pi }{2}⩽\frac{3\pi }{2}$$$$\left(a\right)$$$$\cos (2\theta -\frac{\pi }{2})=\frac{1}{2}$$$$\Rightarrow 2\theta -\frac{\pi }{2}=-\frac{\pi }{3},\frac{\pi }{3}$$$$\Rightarrow \theta =\frac{1}{2}(\frac{\pi }{2}-\frac{\pi }{3})=\frac{\pi }{12}$$$$\Rightarrow \theta =\frac{1}{2}(\frac{\pi }{2}+\frac{\pi }{3})=\frac{5\pi }{12}$$$$\left(b\right)$$$$\cos \theta -\sqrt{3}sin\theta =0$$$$\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}$$$$\Rightarrow \theta =\frac{\pi }{6}$$

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