solve-for-0-pi-the-equations-a-cos-2-pi-2-1-2-b-cos-3-sin-0-

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

s o l v e f o r 0 ⩽ θ ⩽ π, t h e e q u a t i o n s

{a)}_{} \cos (2 θ - \frac{π}{2}) = \frac{1}{2}

b) \cos θ - \sqrt{3} s i n θ = 0

Answered by MrW3 last updated on 30/Jun/18

0 ⩽ θ ⩽ π

- \frac{π}{2} ⩽ 2 θ - \frac{π}{2} ⩽ \frac{3 π}{2}

(a)

\cos (2 θ - \frac{π}{2}) = \frac{1}{2}

\Rightarrow 2 θ - \frac{π}{2} = - \frac{π}{3}, \frac{π}{3}

\Rightarrow θ = \frac{1}{2} (\frac{π}{2} - \frac{π}{3}) = \frac{π}{12}

\Rightarrow θ = \frac{1}{2} (\frac{π}{2} + \frac{π}{3}) = \frac{5 π}{12}

(b)

\cos θ - \sqrt{3} s i n θ = 0

\Rightarrow \tan θ = \frac{1}{\sqrt{3}}

\Rightarrow θ = \frac{π}{6}

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