Question

Solve the following equations
How many roots of the equation $co{s}^{2}x+\frac{\sqrt{3}+1}{2}sinx-\frac{\sqrt{3}}{4}-1=0$ lie in the interval $[-\pi ,\pi ]?$

Answer 1

${\cos }^{2}x+\left(\frac{\sqrt{3}+1}{2}\right)sinx-(\frac{\sqrt{3}}{4}+1)=0$

$\Rightarrow {\sin }^{2}x-\left(\frac{\sqrt{3}+1}{2}\right)sinx+\frac{\sqrt{3}}{4}=0$

$sinx=\frac{\left(\frac{\sqrt{3}+1}{2}\right)\pm \sqrt{\frac{1}{4}(3+1+2\sqrt{3})-4\left(\frac{\sqrt{3}}{4}\right)}}{2}$

$=[\left(\frac{\sqrt{3}+1}{2}\right)\pm \sqrt{1+\frac{\sqrt{3}}{2}-\sqrt{3}}]\frac{1}{2}$

$=\frac{1}{2}[\left(\frac{\sqrt{3}+1}{2}\right)\pm \sqrt{\frac{1-\sqrt{3}}{2}}]$

$=0.866,0.5$

$x=\left(\frac{\pi }{3}/\frac{26}{5}\right);\left(\frac{\pi }{6}/\frac{56}{6}\right)$

Total $4$ roots in $[-\pi ,\pi ]$.