First slide

Trigonometric equations

Question

$Number of solutions of the equation \cos^{2} x + (\frac{\sqrt{3} + 1}{2}) \sin x = \frac{\sqrt{3}}{4} + 1 in the interval [- π, π]$ is

Moderate

A

2
B

4
C

6
D

8

Solution

$Given that \cos^{2} x + (\frac{\sqrt{3} + 1}{2}) \sin x = \frac{\sqrt{3}}{4} + 1$
$\Rightarrow (1 - \sin^{2} x) + (\frac{\sqrt{3} + 1}{2}) \sin x - \frac{\sqrt{3}}{4} - 1 = 0$
$\Rightarrow \sin^{2} x - (\frac{\sqrt{3} + 1}{2}) \sin x + \frac{\sqrt{3}}{4} = 0$
$∴ on solving, we get \sin x = \frac{1}{2}, \frac{\sqrt{3}}{2}$
$∴ x \in [- π, π] are \frac{π}{6}, \frac{5 π}{6}, \frac{π}{3}, \frac{2 π}{3}$
$Number of solutions are 4$

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