\begin{align} \text{Length of } OP & = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ & = \sqrt{ (-1 - 0)^2 + (-\sqrt{3} - 0)^2} \\ & = 2 \text{ units} \end{align}

\begin{align} \sin 240^\circ & = {Opp \over Hyp} \\ & = {-\sqrt{3} \over 2} \\ & = -{\sqrt{3} \over 2} \\ \\ \cos 240^\circ & = {Adj \over Hyp} \\ & = {-1 \over 2} \\ & = -{1 \over 2} \\ \\ \tan 240^\circ & = {Opp \over Adj} \\ & = {-\sqrt{3} \over -1} \\ & = \sqrt{3} \\ \\ \sec 240^\circ & = {1 \over \cos 240^\circ} \\ & = {1 \over -{1 \over 2}} \\ & = -2 \\ \\ \text{cosec } 240^\circ & = {1 \over \sin 240^\circ} \\ & = {1 \over -{\sqrt{3} \over 2}} \\ & = -{2 \over \sqrt{3}} \\ \\ \cot 240^\circ & = {1 \over \tan 240^\circ} \\ & = {1 \over \sqrt{3}} \end{align}

\begin{align} \cos 330^\circ & = \cos 30^\circ \phantom{00000000} [\alpha = 360^\circ - 330^\circ = 30^\circ] \\ & = {\sqrt{3} \over 2} \end{align}

\begin{align} \sin 225^\circ & = -\sin 45^\circ \phantom{00000000} [\alpha = 225^\circ - 180^\circ = 45^\circ] \\ & = -{1 \over \sqrt{2}} \end{align}

\begin{align} \sin {5\pi \over 6} & = \sin {\pi \over 6} \phantom{00000000} \left[ \alpha = \pi - {5\pi \over 6} = {\pi \over 6} \right]\\ & = {1 \over 2} \end{align}

\begin{align} \tan (-45^\circ) & = -\tan 45^\circ \\ & = -1 \end{align}

\begin{align} \cos (-150^\circ) & = -\cos 30^\circ \phantom{00000000} [\alpha = 180^\circ - 150^\circ = 30^\circ] \\ & = - {\sqrt{3} \over 2} \end{align}

\begin{align} \tan \left(-{2\pi \over 3}\right) & = \tan {\pi \over 3} \phantom{00000000} \left[ \alpha = \pi - {2\pi \over 3} = {\pi \over 3}\right] \\ & = \sqrt{3} \end{align}

\begin{align} \sin \theta & = -0.2 \phantom{000000} [\text{3rd or 4th quadrant since } \sin \theta < 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.2) \\ & = 11.54^\circ \end{align}

\begin{align} \theta & = 180^\circ + 11.54^\circ, 360^\circ - 11.54^\circ \\ & = 191.54^\circ, 348.46^\circ \\ & \approx 191.5^\circ, 348.5^\circ \end{align}

\begin{align} \tan \theta & = -3 \phantom{000000} [\text{2nd or 4th quadrant since } \tan \theta < 0]\\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (3) \\ & = 71.57^\circ \end{align}

\begin{align} \theta & = 180^\circ - 71.57^\circ, 360^\circ - 71.57^\circ \\ & = 108.43^\circ, 288.43^\circ \\ & \approx 108.4^\circ, 288.4^\circ \end{align}

\begin{align} \cos \theta & = {1 \over \sqrt{2}} \phantom{000000} [\text{1st & 4th quadrant since } \cos \theta > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over \sqrt{2}\right) \\ & = {\pi \over 4} \end{align}

\begin{align} \theta & = {\pi \over 4}, 2\pi - {\pi \over 4} \\ & = {\pi \over 4}, {7\pi \over 4} \end{align}

\begin{align} \cos \theta & = -0.7 \phantom{000000} [\text{2nd & 3rd quadrant since } \cos \theta < 0]\\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.7) \\ & = 0.79539 \end{align}

\begin{align} \theta & = \pi - 0.79539, \pi + 0.79539 \\ & = 2.3462, 3.9369 \\ & \approx 2.35, 3.94 \end{align}

\begin{align} \tan^2 \theta & = 3 \\ \tan \theta & = \pm \sqrt{3} \phantom{000000} [\text{All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (\sqrt{3}) \\ & = 60^\circ \end{align}

\begin{align} \theta & = 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} 3\sin^2 \theta & = 2 \\ \sin^2 \theta & = {2 \over 3} \\ \sin \theta & = \pm \sqrt{2 \over 3} \phantom{000000} [\text{All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(\sqrt{2 \over 3}\right) \\ & = 0.95531 \end{align}

\begin{align} \theta & = 0.95531, \pi - 0.95531, \pi + 0.95531, 2\pi - 0.95531 \\ & = 0.95531, 2.1862, 4.0969, 5.3278 \\ & \approx 0.955, 2.19, 4.10, 5.33 \end{align}

\begin{align} \tan A & = -{8 \over 15} \\ {Opp \over Adj} & = {8 \over -15} \end{align}

\begin{align} x & = \sqrt{(8)^2 + (15)^2} \\ & = 17 \\ \\ \cos A & = {Adj \over Hyp} \\ & = {-15 \over 17} \\ & = -{15 \over 17} \\ \\ \sin A & = {Opp \over Hyp} \\ & = {8 \over 17} \end{align}

\begin{align} \sin B & = -{2 \over \sqrt{5}} \phantom{000000} [\text{3rd quadrant since } \sin B < 0] \\ {Opp \over Hyp} & = {-2 \over \sqrt{5}} \end{align}

\begin{align} x & = \sqrt{ (\sqrt{5})^2 - (2)^2 } \\ & = 1 \\ \\ \cos B & = {Adj \over Hyp} \\ & = {-1 \over \sqrt{5}} \\ & = -{1 \over \sqrt{5}} \\ \\ \cot B & = {1 \over \tan B} \\ & = {1 \over {Opp \over Adj}} \\ & = {1 \over {-2 \over -1}} \\ & = {1 \over 2} \end{align}

\begin{align} \cos A & > 0 \text{ and } \sin A > 0 \\ \\ \implies A & \text{ is in 1st quadrant} \\ \\ \cos A & = {1 \over 2} \\ {Adj \over Hyp} & = {1 \over 2} \end{align}

\begin{align} x & = \sqrt{(2)^2 - (1)^2} \\ & = \sqrt{3} \\ \\ \sin (-A) & = -\sin A \\ & = -{Opp \over Hyp} \\ & = -{\sqrt{3} \over 2} \\ \\ \tan A & = {Opp \over Adj} \\ & = {\sqrt{3} \over 1} \\ & = \sqrt{3} \end{align}

\begin{align} \tan A & < 0 \text{ and } \cos A > 0 \\ \\ \implies A & \text{ lies in 4th quadrant} \\ \\ \tan A & = -{5 \over 12} \\ {Opp \over Adj} & = {-5 \over 12} \end{align}

\begin{align} x & = \pm \sqrt{(5)^2 + (12)^2} \\ & = 13 \\ \\ \cos (-A) & = \cos A \\ & = {Adj \over Hyp} \\ & = {12 \over 13} \end{align}

\begin{align} \sin 20^\circ & = k \\ {Opp \over Hyp} & = {k \over 1} \end{align}

\begin{align} x & = \sqrt{ (1)^2 - (k)^2 } \\ & = \sqrt{1 - k^2} \\ \\ \sin 200^\circ & = \sin (180^\circ + 20^\circ) \phantom{000000} [\text{Third quadrant, } \sin \theta < 0] \\ & = -\sin 20^\circ \\ & = -k \end{align}

\begin{align} 100 + 71\sec x & = 0 \\ 71\sec x & = -100 \\ 71 \left(1 \over \cos x\right) & = -100 \\ {1 \over \cos x} & = -{100 \over 71} \\ \\ \cos x & = -{71 \over 100} \\ \cos x & = -0.71 \phantom{000000} [\text{2nd or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.71) \\ & = 44.77^\circ \end{align}

\begin{align} x & = 180^\circ - 44.77^\circ, 180^\circ + 44.77^\circ \\ & = 135.23^\circ, 224.77^\circ \\ & \approx 135.2^\circ, 244.8^\circ \end{align}

\begin{align} 5 \cot x + 3 & = 2 + 3\cot x \\ 5 \cos x - 3\cot x & = 2 - 3 \\ 2\cot x & = -1 \\ \cot x & = -{1 \over 2} \\ {1 \over \tan x} & = -{1 \over 2} \\ 2 & = - \tan x \\ \tan x & = -2 \phantom{000000} [\text{2nd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2) \\ & = 63.43^\circ \end{align}

\begin{align} x & = 180^\circ - 63.43^\circ, 360^\circ - 63.43^\circ \\ & = 116.57^\circ, 296.57^\circ \\ & \approx 116.6^\circ, 296.6^\circ \end{align}

\begin{align} 5 \cos x & = \sqrt{3} \sin 60^\circ \\ 5 \cos x & = \sqrt{3} \left(\sqrt{3} \over 2\right) \\ 5 \cos x & = {3 \over 2} \\ \cos x & = {3 \over 2} \div 5 \\ \cos x & = {3 \over 10} \\ \cos x & = 0.3 \phantom{00000000} [\text{1st or 4th quadrant}]\\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.3) \\ & = 72.54^\circ \end{align}

\begin{align} x & = 72.54^\circ, 360^\circ - 72.54^\circ \\ & = 72.54^\circ, 287.46^\circ \\ & \approx 72.5^\circ, 287.5^\circ \end{align}

\begin{align} 2 \sin (-x) & = 0.3 \\ \sin (-x) & = 0.3 \div 2 \\ \sin (-x) & = 0.15 \\ -\sin x & = 0.15 \phantom{-00000000} [\sin (-A) = -\sin A] \\ \sin x & = -0.15 \phantom{00000000} [\text{3rd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.15) \\ & = 8.63^\circ \end{align}

\begin{align} x & = 180^\circ + 8.63^\circ, 360^\circ - 8.63^\circ \\ & = 188.63^\circ, 351.37^\circ \\ & \approx 188.6^\circ, 351.4^\circ \end{align}

\begin{align} 2 \cos x & = \sec x \\ 2 \cos x & = {1 \over \cos x} \\ 2 \cos^2 x & = 1\\ \cos^2 x & = {1 \over 2} \\ \cos x & = \pm \sqrt{1 \over 2} \phantom{00000000} [\text{All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(\sqrt{1 \over 2}\right) \\ & = 45^\circ \end{align}

\begin{align} x & = 45^\circ, 180^\circ - 45^\circ, 180^\circ + 45^\circ, 360^\circ - 45^\circ \\ & = 45^\circ, 135^\circ, 225^\circ, 315^\circ \end{align}

\begin{align} 3 \sin x + 2 & = \cot 15^\circ \\ 3 \sin x + 2 & = {1 \over \tan 15^\circ} \\ 3 \sin x + 2 & = 3.7321 \\ 3 \sin x & = 3.7321 - 2 \\ 3 \sin x & = 1.7321 \\ \sin x & = 1.7321 \div 3 \\ \sin x & = 0.57736 \phantom{00000000} [\text{1st or 2nd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.57736) \\ & = 35.27^\circ \end{align}

\begin{align} x & = 35.27^\circ, 180^\circ - 35.27^\circ \\ & = 35.27^\circ, 144.73^\circ \\ & \approx 35.3^\circ, 144.7^\circ \end{align}

\begin{align} 2\sin x - \sqrt{3} & = 0 \\ 2\sin x & = \sqrt{3} \\ \sin x & = {\sqrt{3} \over 2} \phantom{000000} \text{[1st or 2nd quadrants}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(\sqrt{3} \over 2\right) \phantom{0000} [\text{Radian mode!}] \\ & = {\pi \over 3} \end{align}

\begin{align} x & = {\pi \over 3}, \pi - {\pi \over 3} \\ & = {\pi \over 3}, {2\pi \over 3} \end{align}

\begin{align} {\cos x - 1 \over 0.5 - \cos x} & = 2 \\ \cos x - 1 & = 2(0.5 - \cos x) \\ \cos x - 1 & = 1 - 2\cos x \\ 3\cos x & = 2 \\ \cos x & = {2 \over 3} \phantom{000000} [\text{1st or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(2 \over 3\right) \phantom{000} [\text{Radian mode!}] \\ & = 0.84107 \end{align}

\begin{align} x & = 0.84107, 2\pi - 0.84107 \\ & = 0.84107, 5.4421 \\ & \approx 0.841, 5.44 \end{align}

\begin{align} {4 \over 1 - 2\tan^2 x} & = 7 \\ 4 & = 7(1 - 2\tan^2 x) \\ 4 & = 7 - 14 \tan^2 x \\ 14\tan^2 x & = 7 - 4 \\ 14\tan^2 x & = 3 \\ \tan^2 x & = {3 \over 14} \\ \tan x & = \pm \sqrt{3 \over 14} \phantom{000000000} \text{[All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(\sqrt{3 \over 14}\right) \phantom{0000} [\text{Radian mode!}] \\ & = 0.43354 \end{align}

\begin{align} x & = 0.43354, \pi - 0.43354, \pi + 0.43354, 2\pi - 0.43354 \\ & = 0.43352, 2.7081, 3.5751, 5.8496 \\ & \approx 0.434, 2.71, 3.58, 5.85 \end{align}

\begin{align} \require{cancel} \sqrt{3} \sin \left(x - {\pi \over 2}\right) + 3 & = \sqrt{3} \cos x \\ \\ \text{Since } \sin \left(x - {\pi \over 2}\right) & = - \cos x, \\ \sqrt{3} (-\cos x) + 3 & = \sqrt{3} \cos x \\ -\sqrt{3} \cos x + 3 & = \sqrt{3} \cos x \\ 3 & = 2\sqrt{3} \cos x \\ \\ \cos x & = {3 \over 2\sqrt{3}} \\ \cos x & = {3 \over 2\sqrt{3}} \times {\sqrt{3} \over \sqrt{3}} \\ \cos x & = {\cancel{3}\sqrt{3} \over 2(\cancel{3})} \\ \cos x & = {\sqrt{3} \over 2} \phantom{00000000} [\text{1st or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(\sqrt{3} \over 2\right) \\ & = {\pi \over 6} \end{align}

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

\begin{align} \tan \theta & = 7 \phantom{000000} [\text{1st or 3rd quadrant}] \\ \\ \text{Since } 0^\circ < & \phantom{.} \theta < 180^\circ, \theta \text{ is in 1st quadrant} \\ \\ \tan \theta & = 7 \\ {Opp \over Adj} & = {7 \over 1} \end{align}

\begin{align} \require{cancel} x & = \sqrt{ (1)^2 + (7)^2 } \\ & = \sqrt{50} \\ & = \sqrt{25} \sqrt{2} \\ & = 5\sqrt{2} \\ \\ \sin \theta & = {Opp \over Hyp} \\ & = {7 \over 5\sqrt{2}} \\ \\ \sin 135^\circ & = \sin (180^\circ - 135^\circ) \\ & = \sin 45^\circ \phantom{000000} [\sin (180^\circ - A) = \sin A] \\ & = {1 \over \sqrt{2}} \\ \\ \therefore v & = {770 \sin 135^\circ \over \sin \theta} \\ & = {770 \left(1 \over \sqrt{2}\right) \over {7 \over 5\sqrt{2}}} \\ & = {770 \over \sqrt{2}} \div {7 \over 5\sqrt{2}} \\ & = {770 \over \cancel{\sqrt{2}}} \times {5\cancel{\sqrt{2}} \over 7} \\ & = {3850 \over 7} \\ & = 550 \end{align}

\begin{align} \tan A & = 2 \phantom{0000000000} [\text{1st or 3rd quadrants}] \\ {Opp \over Adj} & = {2 \over 1} \text{ or } {-2 \over -1} \\ \\ 3 \sin B + 2 & = 0 \\ 3 \sin B & = -2 \\ \sin B & = -{2 \over 3} \phantom{0000000000} [\text{3rd or 4th quadrants}] \\ {Opp \over Hyp} & = {-2 \over 3} \\ \\ \\ \text{Case 1: } & A \text{ in 1st quadrant, } B \text{ in 3rd quadrant} \\ \text{Case 2: } & A \text{ in 1st quadrant, } B \text{ in 4th quadrant} \\ \text{Case 3: } & A \text{ in 3rd quadrant, } B \text{ in 4th quadrant} \\ \end{align}

\begin{align} 3 \sin A - \tan B & = 3 \left(2 \over \sqrt{5}\right) - {-2 \over -\sqrt{5}} \\ & = {6 \over \sqrt{5}} - {2 \over \sqrt{5}} \\ & = {4 \over \sqrt{5}} \end{align}

\begin{align} 3 \sin A - \tan B & = 3 \left(2 \over \sqrt{5}\right) - {-2 \over \sqrt{5}} \\ & = {6 \over \sqrt{5}} + {2 \over \sqrt{5}} \\ & = {8 \over \sqrt{5}} \end{align}

\begin{align} 3 \sin A - \tan B & = 3 \left(-2 \over \sqrt{5}\right) - {-2 \over \sqrt{5}} \\ & = {-6 \over \sqrt{5}} + {2 \over \sqrt{5}} \\ & = -{4 \over \sqrt{5}} \\ \\ \\ \therefore 3 \sin A - \tan B & = {4 \over \sqrt{5}} \text{ or } {8 \over \sqrt{5}} \text{ or } -{4 \over \sqrt{5}} \end{align}

\begin{align} \sin 50^\circ & = k \\ {Opp \over Adj} & = {k \over 1} \end{align}

\begin{align} \tan 40^\circ & = {\sqrt{1 - k^2} \over k} \end{align}

\begin{align} {3 + \sqrt{10} \cos x \over \sqrt{10} \sin x + 1} & = 0 \\ \\ 3 + \sqrt{10} \cos x & = 0 \\ \sqrt{10} \cos x & = -3 \\ \cos x & = -{3 \over \sqrt{10}} \phantom{000000} [\text{2nd or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(3 \over \sqrt{10}\right) \\ & = 18.43^\circ \end{align}

\begin{align} x & = 180^\circ - 18.43^\circ, 180^\circ + 18.43^\circ \\ & = 161.57^\circ, 198.43^\circ \\ \\ \text{For } & x = 161.57^\circ, \\ \sqrt{10} \sin 161.57^\circ + 1 & = 1.999 \phantom{.} 740 \\ & \approx 2 \\ \\ \text{For } & x = 198.43^\circ, \\ \sqrt{10} \sin 198.43^\circ + 1 & = 0.000 \phantom{.} 259 \\ & \approx 0 \\ \\ \text{Reject } x = 198.43^\circ & \text{ since equation would be undefined} \\ \\ \\ \therefore x & \approx 161.6^\circ \end{align}

\begin{align} 3 \sin x - 2 & = 0 \\ 3 \sin x & = 2 \\ \sin x & = {2 \over 3} \phantom{000000} [\text{1st or 2nd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(2 \over 3\right) \\ & = 41.81^\circ \end{align}

\begin{align} x & = 41.81^\circ, 180^\circ - 41.81^\circ \\ & = 41.81^\circ, 138.19^\circ \\ & \approx 41.8^\circ, 138.2^\circ \end{align}

\begin{align} \tan x - 4 & = 0 \\ \tan x & = 4 \phantom{000000} [\text{1st or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (4) \\ & = 75.96^\circ \end{align}

\begin{align} x & = 75.96^\circ, 180^\circ + 75.96^\circ \\ & = 75.96^\circ, 255.96^\circ \\ & \approx 76.0^\circ, 256.0^\circ \end{align}