Q.Solve the equation: $\sin x + \sqrt{3} \cos x = \sqrt{2}$

While the principal solutions capture the answers within one period, the general solution is a single formula, parametrised by an arbitrary integer $n$, that generates every solution of a trigonometric equation across all of $\mathbb{R}$. The three foundational results are: the general solution of $\sin\theta=\sin\alpha$ is $\theta=n\pi+(-1)^n\alpha$; of $\cos\theta=\cos\alpha$ is $\theta=2n\pi\pm\alpha$; and of $\tan\theta=\tan\alpha$ is $\theta=n\pi+\alpha$, each for $n\in\mathbb{Z}$. These are proved by combining the allied-angle identity for each function with the function's own periodicity. Squared versions follow immediately: $\sin^2\theta=\sin^2\alpha$, $\cos^2\theta=\cos^2\alpha$, and $\tan^2\theta=\tan^2\alpha$ all share the single general solution $\theta=n\pi+\alpha$. In practice, solving a general trigonometric equation means algebraically manipulating it (factorising, using double-angle or sum-to-product identities, or squaring and checking) until it is reduced to one of these three basic recognisable forms with a standard angle on the right, then applying the matching formula. …