\begin{align*} \cos A & = {4 \over 5} \phantom{000000} [\text{1st or 4th quadrant}] \\ \\ \sin B & = -{1 \over 2} \phantom{000000} [\text{3rd or 4th quadrant}] \\ \\ \therefore A & \text{ and } B \text{ lies in 4th quadrant} \\ \\ \cos A & ={4 \over 5} \\ {\text{Adj} \over \text{Hyp}} & = {4 \over 5} \end{align*}

\begin{align*} \sqrt{ 5^2 - 4^2 } & = 3 \\ \\ \sin \left(A + {\pi \over 3} \right) & = \sin A \cos {\pi \over 3} + \cos A \sin {\pi \over 3} \\ & = \left(-{3 \over 5}\right)\left(1 \over 2\right) + \left(4 \over 5\right) \left(\sqrt{3} \over 2\right) \phantom{000000} [\text{Special angle } {\pi \over 3} = 60^\circ ] \\ & = -{3 \over 10} + {4 \sqrt{3} \over 10} \\ & = -{3 \over 10} + {2 \over 5} \sqrt{3} \end{align*}

\begin{align*} \sin B & = -{1 \over 2} \\ {\text{Opp} \over \text{Hyp}} & = {-1 \over 2} \end{align*}

\begin{align*} \sqrt{2^2 - 1^2} & = \sqrt{3} \\ \\ \tan (A + B) & = { \tan A + \tan B \over 1 - \tan A \tan B } \\ & = { - {3 \over 4} + \left(-{1 \over \sqrt{3}}\right) \over 1 - \left(-{3 \over 4}\right) \left(-{1 \over \sqrt{3}}\right) } \\ & = { - {3 \over 4} - {1 \over \sqrt{3}} \over 1 - {3 \over 4 \sqrt{3}} } \times {4 \sqrt{3} \over 4 \sqrt{3}} \\ & = { - {12 \sqrt{3} \over 4} - {4 \sqrt{3} \over \sqrt{3}} \over 4 \sqrt{3} - {12 \sqrt{3} \over 4 \sqrt{3}} } \\ & = { - 3 \sqrt{3} - 4 \over 4 \sqrt{3} - 3 } \times { 4 \sqrt{3} + 3 \over 4 \sqrt{3} + 3 } \\ & = { - 12(3) - 9 \sqrt{3} - 16 \sqrt{3} - 12 \over (4 \sqrt{3})^2 - 3^2 } \phantom{000000} [ (a + b)(a - b) = a^2 - b^2 ] \\ & = { - 36 - 25 \sqrt{3} - 12 \over 39} \\ & = { -48 - 25 \sqrt{3} \over 39} \\ & = - {48 \over 39} - {25 \sqrt{3} \over 39} \\ & = - {48 \over 39} - {25 \over 39} \sqrt{3} \end{align*}

\begin{align*} \cos (x + 60^\circ) & = \cos x \\ \cos x \cos 60^\circ - \sin x \sin 60^\circ & = \cos x \\ (\cos x)\left(1 \over 2\right) - (\sin x) \left(\sqrt{3} \over 2\right) & = \cos x \\ {1 \over 2} \cos x - {\sqrt{3} \over 2} \sin x & = \cos x \\ -{\sqrt{3} \over 2} \sin x & = \cos x - {1 \over 2} \cos x \\ -{\sqrt{3} \over 2} \sin x & = {1 \over 2} \cos x \\ - \sqrt{3} \sin x & = \cos x \\ - {\sqrt{3} \sin x \over \cos x} & = 1 \\ - \sqrt{3} \tan x & = 1 \\ \tan x & = -{1 \over \sqrt{3}} \phantom{0000000} [\text{2nd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over \sqrt{3}\right) \\ & = 30^\circ \end{align*}

\begin{align*} x & = 180^\circ - 30^\circ, 360^\circ - 30^\circ \\ & = 150^\circ, 330^\circ \end{align*}

\begin{align*} 4 \sin 3x \cos x & = 4 \cos 3x \sin x + 3 \\ 4 \sin 3x \cos x - 4 \cos 3x \sin x & = 3 \\ 4( \sin 3x \cos x - \cos 3x \sin x) & = 3 \\ \sin 3x \cos x - \cos 3x \sin x & = {3 \over 4} \\ \sin (3x - x) & = {3 \over 4} \phantom{0000000000} [\sin (A - B) = \sin A \cos B - \cos A \sin B] \\ \sin 2x & = {3 \over 4} \phantom{0000000000} [\text{1st or 2nd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(3 \over 4\right) \\ & = 48.59^\circ \end{align*}

\begin{align*} [ \text{Since } & 0^\circ \le x \le 360^\circ, 0^\circ \le 2x \le 720^\circ ] \\ \\ 2x & = 48.59^\circ, 180^\circ - 48.59^\circ \\ & = 48.59^\circ, 131.41^\circ, 48.59^\circ + 360^\circ, 131.41^\circ + 360^\circ \\ & = 48.59^\circ, 131.41^\circ, 408.59^\circ, 491.41^\circ \\ \\ x & = 24.295^\circ, 65.705^\circ, 204.295^\circ, 245.705^\circ \\ & \approx 24.3^\circ, 65.7^\circ, 204.3^\circ, 245.7^\circ \end{align*}