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Question Number 8273 by lepan last updated on 05/Oct/16

E x p r e s s s i n α + \sqrt{3} c o s α i n t h e f o r m

R s i n (α + β) w h e r e R > 0 a n d 0 ° < β < 90 ° .

H e n c e s o l v e t h e e q u a t i o n s i n α + \sqrt{3} c o s α = 2

f o r 0 ° < α < 270 ° .

Answered by prakash jain last updated on 05/Oct/16

\sin α + \sqrt{3} \cos α

= 2 (\frac{1}{2} \sin α + \frac{\sqrt{3}}{2} \cos α)

= 2 (\sin \frac{π}{6} \sin α + \cos \frac{π}{6} \cos α)

= 2 \cos (\frac{π}{6} - α)

\sin α + \sqrt{3} \cos α = 1

\Rightarrow 2 \cos (\frac{π}{6} - α) = 1

\Rightarrow \cos (\frac{π}{6} - α) = \frac{1}{2}

\Rightarrow \cos (\frac{π}{6} - α) = \cos \frac{π}{3}

\frac{π}{6} - α = 2 n π \pm \frac{π}{3}

α = - 2 n π + (\frac{π}{6} \pm \frac{π}{3}) = {\begin{cases} 2 k π + \frac{π}{2} \\ 2 k π - \frac{π}{6} \end{cases}

between 0 to 270 ° (o r 0 to \frac{3 π}{2})

α = \frac{π}{2} or 90 °