First slide

Trigonometric equations

Question

If $x + y = \frac{2 π}{3} a n d \frac{sinx}{\sin y} = 2$ then

Moderate

A

The number of values of $x \in$ $[0, 4 π]$ are 4
B

The number of values of $x \in [0, 4 π]$ are 2
C

The number of values of $y \in [0, 4 π]$ are 4
D

The number of values of $y \in [0, 4 π]$ are 8

Solution

$\begin{array}{l} x + y = \frac{2 π}{3} \Rightarrow y = \frac{2 π}{3} - x \\ ∴ \sin x = 2 \sin (\frac{2 π}{3} - x) \\ = 2 [(\frac{\sqrt{3}}{2}) \cos x + \frac{1}{2} \sin x] \\ = \sqrt{3} \cos x + \sin x \\ \Rightarrow \cos x = 0 \Rightarrow x = n π + \frac{π}{2} \\ \Rightarrow y = \frac{2 π}{3} - n π - \frac{π}{2} = \frac{π}{6} - n π \\ H e n c e, \\ x \in [0, 4 π], x = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2} \\ a n d f o r x \in [0, 4 π], y = \frac{π}{6}, \frac{7 π}{6}, \frac{13 π}{6}, \frac{19 π}{6} \end{array}$

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