Question

Number of roots of the equation $cos^{2} x + \frac{\sqrt{3} + 1}{2} sin ? x - \frac{\sqrt{3}}{4} - 1 = 0$ which lie in the interval $\left[- \pi , \pi \right]$ is

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (B)

4

Answer 2

Given equation is $1-sin^{2} x + \frac{\sqrt{3} + 1}{2} sin ? x - \frac{\sqrt{3}}{4} - 1 = 0$ $? \, sin^{2} x - \frac{\sqrt{3} + 1}{2} sin ? x + \frac{\sqrt{3}}{4} = 0 ; 4 sin^{2} ? x - 2 \sqrt{3} sin ? x - 2 sin ? x + \sqrt{3} = 0$ $2sinx\left(\right. 2 sin x - \sqrt{3} \left.\right)-\left(\right. 2 sin x - \sqrt{3} \left.\right)=0$ $\Rightarrow \left(\right. 2 sin x - 1 \left.\right)\left(\right. 2 sin x - \sqrt{3} \left.\right)=0$ On solving we get $sin x=\frac{1}{2}$ $;\frac{\sqrt{3}}{2}$ $x=\frac{\pi }{6},\frac{5 \pi }{6} \, ;\frac{\pi }{3},\frac{2 \pi }{3}$