\begin{align*} \sqrt{3} \tan x + 1 & = 0 \\ \sqrt{3} \tan x & = -1 \\ \tan x & = -{1 \over \sqrt{3}} \phantom{000000} [\text{2nd or 4th quadrant since } \tan x < 0 ] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over \sqrt{3}\right) \\ & = 30^\circ \\ & = {\pi \over 6} \text{ radians} \end{align*}

\begin{align*} x & = \pi - {\pi \over 6}, 2\pi - {\pi \over 6} \\ & = {5\pi \over 6}, {11\pi \over 6} \end{align*}

\begin{align*} 3\sin 2\theta - 1 & = 0 \\ 3\sin 2\theta & = 1 \\ \sin 2\theta & = {1 \over 3} \phantom{000000} [\text{1st or 2nd quadrant since } \sin 2\theta > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 3\right) \\ & = 19.47^\circ \\ \end{align*}

\begin{align*} [ \text{Since } & -180^\circ < \theta < 180^\circ, -360^\circ < 2 \theta < 360^\circ ] \\ \\ 2 \theta & = 19.47^\circ, 180^\circ - 19.47^\circ \\ & = 19.47^\circ, 160.53^\circ, 19.47^\circ - 360^\circ, 160.53^\circ - 360^\circ \\ & = -340.53^\circ, -199.47^\circ, 19.47^\circ, 160.53^\circ \\ \\ \theta & = -170.265^\circ, -99.735^\circ, 9.735^\circ, 80.275^\circ \\ & \approx -170.3^\circ, -99.7^\circ, 9.7^\circ, 80.3^\circ \end{align*}

\begin{align*} \tan^2 x - 3 & = 0 \\ \tan^2 x & = 3 \\ \tan x & = \pm \sqrt{3} \phantom{000000} [\text{All four quadrants}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (\sqrt{3}) \\ & = 60^\circ \end{align*}

\begin{align*} x & = 60^\circ, 180^\circ - 60^\circ, 180^\circ + 60^\circ, 360^\circ - 60^\circ \\ & = 60^\circ, 120^\circ, 240^\circ, 300^\circ \end{align*}

\begin{align*} 4 (\sin 2x - \cos 2x) & = \cos 2x \\ 4 \sin 2x - 4 \cos 2x & = \cos 2x \\ 4 \sin 2x & = 5 \cos 2x \\ {4 \sin 2x \over \cos 2x} & = 5 \\ 4 \tan 2x & = 5 \\ \tan 2x & = {5 \over 4} \phantom{000000} [\text{1st or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(5 \over 4\right) \\ & = 51.34^\circ \end{align*}

\begin{align*} [ \text{Since } & 0^\circ < x < 360^\circ, 0^\circ < 2x < 720^\circ ] \\ \\ 2x & = 51.34^\circ, 180^\circ + 51.34^\circ \\ & = 51.34^\circ, 231.34^\circ, 51.34^\circ + 360^\circ, 231.34^\circ + 360^\circ \\ & = 51.34^\circ, 231.34^\circ, 411.34^\circ, 591.34^\circ \\ \\ x & = 25.67^\circ, 115.67^\circ, 205.67^\circ, 295.67^\circ \\ & \approx 25.7^\circ, 115.7^\circ, 205.7^\circ, 295.7^\circ \end{align*}

\begin{align*} 3 \sin x & = 2 \sin x \cos x \\ 3 \sin x - 2 \sin x \cos x & = 0 \\ \sin x (3 - 2 \cos x) & = 0 \\ \\ \sin x = 0 \phantom{0} \text{ or } & \phantom{0} 3 - 2 \cos x = 0 \\ & \phantom{00} -2 \cos x = -3 \\ & \phantom{00000.} \cos x = {-3 \over -2} \\ & \phantom{00000.} \cos x = 1.5 \phantom{0} \text{ (No solutions since } -1 \le \cos x \le 1) \\ \\ \\ \sin x & = 0 \end{align*}

\begin{align*} x & = 0 \text{ (N.A.)}, \pi, 2\pi \text{ (N.A.)} \\ \\ \therefore x & = \pi \end{align*}

\begin{align*} 2 \cos^2 x & = 5 - 4\cos x \\ 2 \cos^2 x + 4 \cos x - 5 & = 0 \phantom{0000000000000000000} [\text{Quadratic equation in } \cos x] \\ \\ \cos x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \phantom{000000} [\text{Use this since equation cannot be factorised}] \\ & = {-4 \pm \sqrt{(4)^2 - 4(2)(-5)} \over 2(2)} \\ & = {-4 \pm \sqrt{56} \over 4} \\ & = 0.8708 \text{ or } -2.8708 \text{ (No solutions since } -1 \le \cos x \le 1) \\ \\ \cos x & = 0.8708 \phantom{000000} [\text{1st or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.8708) \\ & = 0.5139 \end{align*}

\begin{align*} x & = 0.5139, 2\pi - 0.5139 \\ & = 0.5139, 5.7692 \\ & \approx 0.514, 5.77 \end{align*}