Q.Find the principal solution of $\cos\theta = \dfrac{-1}{2}$.

Trigonometric Equations

A trigonometric equation is solved by reducing it to a basic form and writing its

general solution:

$\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^n\alpha$;

$\cos\theta=\cos\alpha\Rightarrow\theta=2n\pi\pm\alpha$;

$\tan\theta=\tan\alpha\Rightarrow\theta=n\pi+\alpha$, with $n\in\mathbb Z$.

Strategy: use identities (double/half-angle, $\sin^2+\cos^2=1$, product-to-sum) to write

everything in one ratio or to factor the equation into simpler pieces; solve each

piece; then discard values outside the required interval or those that violate a

stated restriction (such as $\sin\theta\neq0$). Counting the number of solutions in a

window like $(-2\pi,2\pi)$ is done by listing the admissible $n$. Equations disguised as

a geometric-progression or algebraic condition on $\sin\theta,\cos\theta,\cot\theta$ …