1. We are asked to solve the following trigonometric equations for $x$: (i) $\sin x = \frac{1}{2}$ (ii) $\sin x = -\frac{\sqrt{2}}{2}$ (iii) $\sin x = 0$ (iv) $\sin x = \frac{1}{3}$ (v) $\sin x = -\frac{1}{4}$ (vi) $\cos x = \frac{1}{2}$ (vii) $\cos x = -\frac{\sqrt{3}}{2}$ (viii) $\cos x = -1$ (ix) $\cos x = \frac{1}{3}$ (x) $\cos x = -\frac{1}{4}$ (xi) $\tan x = \sqrt{3}$ (xii) $\tan x = 1$ (xiii) $\tan x = 2$ 2. The general solutions for sine, cosine, and tangent equations are: - For $\sin x = a$, solutions are $x = \arcsin(a) + 2k\pi$ or $x = \pi - \arcsin(a) + 2k\pi$, where $k$ is any integer. - For $\cos x = a$, solutions are $x = \arccos(a) + 2k\pi$ or $x = -\arccos(a) + 2k\pi$. - For $\tan x = a$, solutions are $x = \arctan(a) + k\pi$. 3. Now we solve each: (i) $\sin x = \frac{1}{2}$ $\Rightarrow x = \frac{\pi}{6} + 2k\pi$ or $x = \frac{5\pi}{6} + 2k\pi$ (ii) $\sin x = -\frac{\sqrt{2}}{2}$ $\Rightarrow x = -\frac{\pi}{4} + 2k\pi$ or $x = \pi + \frac{\pi}{4} + 2k\pi = \frac{5\pi}{4} + 2k\pi$ (iii) $\sin x = 0$ $\Rightarrow x = k\pi$ (iv) $\sin x = \frac{1}{3}$ $\Rightarrow x = \arcsin(\frac{1}{3}) + 2k\pi$ or $x = \pi - \arcsin(\frac{1}{3}) + 2k\pi$ (v) $\sin x = -\frac{1}{4}$ $\Rightarrow x = -\arcsin(\frac{1}{4}) + 2k\pi$ or $x = \pi + \arcsin(\frac{1}{4}) + 2k\pi$ (vi) $\cos x = \frac{1}{2}$ $\Rightarrow x = \frac{\pi}{3} + 2k\pi$ or $x = -\frac{\pi}{3} + 2k\pi = \frac{5\pi}{3} + 2k\pi$ (vii) $\cos x = -\frac{\sqrt{3}}{2}$ $\Rightarrow x = \frac{5\pi}{6} + 2k\pi$ or $x = \frac{7\pi}{6} + 2k\pi$ (viii) $\cos x = -1$ $\Rightarrow x = \pi + 2k\pi$ (ix) $\cos x = \frac{1}{3}$ $\Rightarrow x = \arccos(\frac{1}{3}) + 2k\pi$ or $x = -\arccos(\frac{1}{3}) + 2k\pi$ (x) $\cos x = -\frac{1}{4}$ $\Rightarrow x = \arccos(-\frac{1}{4}) + 2k\pi$ or $x = -\arccos(-\frac{1}{4}) + 2k\pi$ (xi) $\tan x = \sqrt{3}$ $\Rightarrow x = \frac{\pi}{3} + k\pi$ (xii) $\tan x = 1$ $\Rightarrow x = \frac{\pi}{4} + k\pi$ (xiii) $\tan x = 2$ $\Rightarrow x = \arctan(2) + k\pi$ 4. Here $k$ is any integer representing the periodic nature of trigonometric functions. This completes the solutions for all given equations.