Question

Solve $2{\cos }^2\theta -\sqrt{3}\sin \theta +1=0$

Answer 1

Consider the given equation,

$2{\cos }^2\theta -\sqrt{3}\sin \theta +1=0$

$2(1-{\sin }^2\theta )-\sqrt{3}\sin \theta +1=0$ $[\because {\cos }^2\theta =1-{\sin }^2\theta ]$

$2-2{\sin }^2\theta -\sqrt{3}\sin \theta +1=0$

$2{\sin }^2\theta +\sqrt{3}\sin \theta -3=0$

$\sin \theta =\frac{-\sqrt{3}\pm \sqrt{3-4\times 2\times -3}}{2\times 2}$ (use quadratic formula)

$\sin \theta =\frac{-\sqrt{3}\pm \sqrt{3+24}}{2\times 2}$

$\sin \theta =\frac{-\sqrt{3}\pm 3\sqrt{3}}{4}$

$\therefore \sin \theta =\frac{\sqrt{3}}{2},-\sqrt{3}$