\begin{align} 5 \cos x + 2\sin x & = 0 \\ 2\sin x & = - 5 \cos x \\ {2\sin x \over \cos x} & = -5 \\ {\sin x \over \cos x} & = -{5 \over 2} \\ \tan x & = -2.5 \phantom{00000} [\text{2nd & 4th quadrants since } \tan x < 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2.5) \\ & = 68.198^\circ \end{align}

\begin{align} x & = 180^\circ - 68.198^\circ, 360^\circ - 68.198^\circ \\ & = 111.802^\circ, 291.802^\circ \\ & \approx 111.8^\circ, 291.8^\circ \end{align}

\begin{align} 3(\sin x - \cos x) & = \cos x \\ 3\sin x - 3\cos x & = \cos x \\ 3\sin x & = 4\cos x \\ {3\sin x \over \cos x} & = 4 \\ {\sin x \over \cos x} & = {4 \over 3} \\ \tan x & = {4 \over 3} \phantom{00000} \left[\text{1st & 3rd quadrants since } \tan x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(4 \over 3\right) \\ & = 53.13^\circ \end{align}

\begin{align} x & = 53.13^\circ, 180^\circ + 53.13^\circ \\ & = 53.13^\circ, 233.13^\circ \\ & \approx 53.1^\circ, 233.1^\circ \end{align}

\begin{align} 3\cos x + 7\sin x & = 2\sin x \\ 7\sin x - 2\sin x & = - 3\cos x \\ 5\sin x & = -3 \cos x \\ {5 \sin x \over \cos x} & = - 3 \\ {\sin x \over \cos x} & = -{3 \over 5} \\ \tan x & = -0.6 \phantom{00000} [\text{2nd & 4th quadrants since } \tan x < 0 ] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (0.6) \\ & = 30.96^\circ \end{align}

\begin{align} x & = 180^\circ - 30.96^\circ, 360^\circ - 30.96^\circ \\ & = 149.04^\circ, 329.04^\circ \\ & \approx 149.0^\circ, 329.0^\circ \end{align}

\begin{align} \sin x - \sin x \tan x & = 0 \\ \sin x(1 - \tan x) & = 0 \\ \\ \sin x = 0 \phantom{000} & \text{or} \phantom{000} 1 - \tan x = 0 \\ \\ \\ \sin x & = 0 \end{align}

\begin{align} x = 0^\circ \text{ (Reject)}, & \phantom{.} 180^\circ, 360^\circ \text{ (Reject)} \\ \\ \\ 1 - \tan x & = 0 \\ -\tan x & = -1 \\ \tan x & = 1 \phantom{00000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (1) \\ & = 45^\circ \end{align}

\begin{align} x & = 45^\circ, 180^\circ + 45^\circ \\ & = 45^\circ, 225^\circ \\ \\ \\ \therefore x & = 45^\circ, 180^\circ, 225^\circ \end{align}

\begin{align} \tan x + 5\tan x \sin x & = 0 \\ \tan x (1 + 5\sin x) & = 0 \\ \\ \tan x = 0 \phantom{000} & \text{or}\phantom{000} 1 + 5\sin x = 0 \\ \\ \\ \tan x & = 0 \end{align}

\begin{align} x = 0^\circ \text{ (Reject)}, & \phantom{.} 180^\circ, 360^\circ \text{ (Reject)} \\ \\ \\ 1 + 5\sin x & = 0 \\ 5\sin x & = -1 \\ \sin x & = -{1 \over 5} \\ \sin x & = -0.2 \phantom{00000} [\text{3rd & 4th quadrants since } \sin x < 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.2) \\ & = 11.54^\circ \end{align}

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

\begin{align} 2\sin^2 x + \sin x - 1 & = 0 \\ (2\sin x - 1)(\sin x + 1) & = 0 \end{align} \begin{align} 2\sin x - 1 & = 0 & \text{ or } \phantom{0} \sin x + 1 & = 0 \\ 2\sin x & = 1 & \sin x & = -1 \end{align}

\begin{align} 2\sin x & = 1 \\ \sin x & = {1 \over 2} \phantom{00000} \left[\text{1st & 2nd quadrants since } \sin x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 2\right) \\ & = 30^\circ \end{align}

\begin{align} x & = 30^\circ, 180^\circ - 30^\circ \\ & = 30^\circ, 150^\circ \\ \\ \\ \sin x & = -1 \end{align}

$$ x = 270^\circ $$
$$\therefore x = 30^\circ, 150^\circ, 270^\circ$$

\begin{align} 2\tan^2 x - 3\tan x - 2 & = 0 \\ (2\tan x + 1)(\tan x - 2) & = 0 \\ \\ 2\tan x + 1 = 0 \phantom{00}&\text{ or } \tan x - 2 = 0 \end{align}

\begin{align} 2\tan x + 1 & = 0 \\ 2\tan x & = -1 \\ \tan x & = -{1 \over 2} \phantom{00000} \left[\text{2nd & 4th quadrants since } \tan x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 2\right) \\ & = 26.57^\circ \end{align}

\begin{align} x & = 180^\circ - 26.57^\circ, 360^\circ -26.57^\circ \\ & = 153.43^\circ, 333.43^\circ \\ & \approx 153.4^\circ, 333.4^\circ \\ \\ \\ \tan x - 2 & = 0 \\ \tan x & = 2 \phantom{00000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2) \\ & = 63.43^\circ \end{align}

\begin{align} x & = 63.43^\circ, 180^\circ + 63.43^\circ \\ & = 63.43^\circ, 243.43^\circ \\ & \approx 63.4^\circ, 243.4^\circ \end{align}
\begin{align} \therefore x & = 63.4^\circ, 153.4^\circ, 243.4^\circ, 333.4^\circ \end{align}

\begin{align} 6\cos^2 x - 5\cos x - 1 & = 0 \\ (6\cos x + 1)(\cos x - 1) & = 0 \\ \\ 6\cos x + 1 = 0 \phantom{00}&\text{or}\phantom{00} \cos x - 1 = 0 \end{align}

\begin{align} 6\cos x + 1 & = 0 \\ 6\cos x & = - 1 \\ \cos x & = -{1 \over 6} \phantom{00000} \left[ \text{2nd & 3rd quadrants since } \cos x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 6\right) \\ & = 80.41^\circ \end{align}

\begin{align} x & = 180^\circ - 80.41^\circ, 180^\circ + 80.41^\circ \\ & = 99.59^\circ, 260.41^\circ \\ & \approx 99.6^\circ, 260.4^\circ \end{align}
\begin{align} \cos x - 1 & = 0 \\ \cos x & = 1 \end{align}

$$ x = 0^\circ, 360^\circ $$

$$ \therefore x = 0^\circ, 99.6^\circ, 260.4^\circ, 360^\circ $$

\begin{align} 3\sec^2 x - 7\sec x + 2 & = 0 \\ (3\sec x - 1)(\sec x - 2) & = 0 \\ \\ 3\sec x - 1 = 0 \phantom{0000.}&\text{or}\phantom{000} \sec x - 2 = 0 \end{align}
\begin{align} 3\sec x - 1 & = 0 \\ 3\sec x & = 1 \\ \sec x & = {1 \over 3} \\ {1 \over \cos x} & = {1 \over 3} \\ \\ \cos x & = 3 \text{ (Rejected, since } \cos x \le 1) \end{align}
\begin{align} \sec x - 2 & = 0 \\ \sec x & = 2 \\ {1 \over \cos x} & = 2 \\ 1 & = 2\cos x \\ \\ \cos x & = {1 \over 2} \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 2\right) \\ & = 60^\circ \end{align}

\begin{align} x & = 60^\circ, 360^\circ - 60^\circ \\ & = 60^\circ, 300^\circ \end{align}

\begin{align} 7\sin^3 x + \sin^2 x & = 0 \\ \sin^2 x (7\sin x + 1) & = 0 \\ \\ \sin^2 x = 0 \phantom{0000}&\text{or}\phantom{00} 7\sin x + 1 = 0 \\ \sin x = \pm\sqrt{0} \phantom{0}&\phantom{or00+1} 7\sin x = - 1 \\ \sin x = 0 \phantom{0000}&\phantom{or00+17} \sin x = -{1 \over 7} \end{align}
$$ \sin x = 0 $$

$$ x = 0^\circ, 180^\circ, 360^\circ $$

\begin{align} \sin x & = -{1 \over 7} \phantom{00000} \left[ \text{3rd & 4th quadrants since } \sin x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 7\right) \\ & = 8.21^\circ \end{align}

\begin{align} x & = 180^\circ + 8.21^\circ, 360^\circ - 8.21^\circ \\ & = 188.21^\circ, 351.79^\circ \\ & \approx 188.2^\circ, 351.8^\circ \end{align}

\begin{align} \therefore x & = 0^\circ, 180^\circ, 188.2^\circ, 351.8^\circ, 360^\circ \end{align}

\begin{align} 4\sin x \cos x - 3\sin x & = 0 \\ \sin x (4\cos x - 3) & = 0 \\ \\ \sin x = 0 \phantom{00}&\text{or}\phantom{00} 4\cos x - 3 = 0 \end{align}

$$ \sin x = 0 $$

$$ x = 0, \pi, 2\pi $$

\begin{align} 4\cos x - 3 & = 0 \\ 4\cos x & = 3 \\ \cos x & = {3 \over 4} \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(3 \over 4\right) \\ & = 0.72273 \end{align}

\begin{align} x & = 0.72273, 2\pi - 0.72273 \\ & = 0.72273, 5.5604 \\ & \approx 0.723, 5.56 \end{align}

\begin{align} \therefore x & = 0, 0.723, \pi, 5.56, 2\pi \end{align}

\begin{align} 3\sin^2 x + \sin x \cos x & = 0 \\ \sin x (3 \sin x + \cos x ) & = 0 \\ \\ \sin x = 0 \phantom{00}&\text{or}\phantom{00} 3\sin x + \cos x = 0 \end{align}

$$ \sin x = 0 $$

$$ x = 0, \pi, 2\pi $$

\begin{align} 3\sin x + \cos x & = 0 \\ 3\sin x & = -\cos x \\ {3\sin x \over \cos x} & = -1 \\ {\sin x \over \cos x} & = -{1 \over 3} \\ \tan x & = -{1 \over 3} \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 3\right) \\ & = 0.3217 \end{align}

\begin{align} x & = \pi - 0.3217, 2\pi - 0.3217 \\ & = 2.819, 5.961 \\ & \approx 2.82, 5.96 \end{align}

\begin{align} \therefore x & = 0, 2.82, \pi, 5.96, 2\pi \end{align}

\begin{align} 2\text{ cosec}^2 x - 7\text{cosec } x - 4 & = 0 \\ (2\text{ cosec } x + 1)(\text{cosec } x - 4) & = 0 \\ \\ 2\text{ cosec } x + 1 = 0 \phantom{00}&\text{or}\phantom{00} \text{cosec } x - 4 = 0 \end{align}

\begin{align} 2\text{ cosec } x + 1 & = 0 \\ 2\text{ cosec } x & = -1 \\ \text{cosec } x & = -{1 \over 2} \\ {1 \over \sin x} & = -{1 \over 2} \\ \\ \sin x & = -2 \text{ (Reject, as } \sin x \ge -1) \end{align}

\begin{align} \text{cosec } x - 4 & = 0 \\ \text{cosec } x & = 4 \\ {1 \over \sin x} & = 4 \\ 1 & = 4\sin x \\ \\ \sin x & = {1 \over 4} \phantom{00000} \left[ \text{1st & 2nd quadrants since } \sin x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 4\right) \\ & = 0.2526 \end{align}

\begin{align} x & = 0.2526, \pi - 0.2526 \\ & = 0.2526, 2.888 \\ & \approx 0.253, 2.89 \end{align}

\begin{align} 2\cos^2 x - \cos x - 1 & = 0 \\ (2\cos x + 1)(\cos x - 1) & = 0 \\ \\ 2\cos x + 1 = 0 \phantom{00}&\text{or}\phantom{00} \cos x - 1 = 0 \\ &\phantom{or00-1} \cos x = 1 \end{align}

\begin{align} 2\cos x + 1 & = 0 \\ 2\cos x & = -1 \\ \cos x & = -{1 \over 2} \phantom{00000} \left[ \text{2nd & 3rd quadrants since } \cos x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 2\right) \\ & = 60^\circ \\ & = {\pi \over 3} \end{align}

\begin{align} x & = \pi - {\pi \over 3}, \pi + {\pi \over 3} \\ & = {2\pi \over 3}, {4\pi \over 3} \end{align}

$$ \cos x = 1 $$

$$ x = 0, 2\pi $$

$$ \therefore x = 0, {2\pi \over 3}, {4\pi \over 3}, 2\pi $$

\begin{align} 2\cos x - 3\cos x \sin^2 x & = 0 \\ \cos x (2 - 3\sin^2 x) & = 0 \\ \\ \cos x = 0 \phantom{00}&\text{or}\phantom{00} 2 - 3\sin^2 x = 0 \end{align}

$$ \cos x = 0 $$

$$ x = {\pi \over 2}, {3\pi \over 2} $$

\begin{align} 2 - 3\sin^2 x & = 0 \\ -3 \sin^2 x & = -2 \\ \sin^2 x & = {-2 \over -3} \\ & = {2 \over 3} \\ \sin x & = \pm \sqrt{2 \over 3} \phantom{00000} [\text{All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(\sqrt{2 \over 3}\right) \\ & = 0.9553 \end{align}

\begin{align} x & = 0.9553, \pi - 0.9553, \pi + 0.9553, 2\pi - 0.9553 \\ & = 0.9553, 2.186, 4.096, 5.327 \\ & \approx 0.955, 2.19, 4.10, 5.33 \end{align}

\begin{align} \therefore x & = 0.955, {\pi \over 2}, 2.19, 4.10, {3\pi \over 2}, 5.33 \end{align}

\begin{align} \cos 2x & = 0.5 \phantom{00000} [\text{1st & 4th quadrants since } \cos 2x > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.5) \\ & = 60^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{0} 0^\circ < 2x < 720^\circ \\ \\ 2x & = 60^\circ, 360^\circ - 60^\circ \\ & = 60^\circ, 300^\circ \\ & = 60^\circ, 300^\circ, 60^\circ + 360^\circ, 300^\circ + 360^\circ \phantom{000} [\text{Since } 0^\circ < 2x < 720^\circ] \\ & = 60^\circ, 300^\circ, 420^\circ, 660^\circ \\ \\ x & = {60^\circ \over 2}, {300^\circ \over 2}, {420^\circ \over 2}, {660^\circ \over 2} \\ & = 30^\circ, 150^\circ, 210^\circ, 330^\circ \end{align}

\begin{align} \tan (x - 60^\circ) & = {1 \over \sqrt{3}} \phantom{00000} \left[ \text{Since 1st & 3rd quadrants since } \tan (x - 60^\circ) > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over \sqrt{3}\right) \\ & = 30^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{0} -60^\circ < x - 60^\circ < 300^\circ \\ \\ x - 60^\circ & = 30^\circ, 180^\circ + 30^\circ \\ & = 30^\circ, 210^\circ \\ \\ x & = 30^\circ + 60^\circ, 210^\circ + 60^\circ \\ & = 90^\circ, 270^\circ \end{align}

\begin{align} 3\sin 2x + 2 & = 0 \\ 3\sin 2x & = - 2 \\ \sin 2x & = -{2 \over 3} \phantom{00000} \left[ \text{3rd & 4th quadrants since } \sin 2x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(2 \over 3\right) \\ & = 41.81^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{0} 0^\circ < 2x < 720^\circ \\ \\ 2x & = 180^\circ + 41.81^\circ, 360^\circ - 41.81^\circ \\ & = 221.81^\circ, 318.19^\circ \\ & = 221.81^\circ, 318.19^\circ, 221.81^\circ + 360^\circ, 318.19^\circ + 360^\circ \phantom{000} [\text{Since } 0^\circ < 2x < 720^\circ ] \\ & = 221.81^\circ, 318.19^\circ, 581.81^\circ, 678.19^\circ \\ \\ x & = 110.905^\circ, 159.095^\circ, 290.905^\circ, 339.095^\circ \\ & \approx 110.9^\circ, 159.1^\circ, 290.9^\circ, 339.1^\circ \end{align}

\begin{align} \cos (2x - 40^\circ) & = 0.8 \phantom{00000} [\text{1st & 4th quadrants since } \cos (2x - 40^\circ) > 0 ] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.8) \\ & = 36.86^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 720^\circ \rightarrow \phantom{.} -40^\circ < 2x - 40^\circ < 680^\circ \\ \\ 2x - 40^\circ & = 36.86^\circ, 360^\circ - 36.86^\circ \\ & = 36.86^\circ, 323.14^\circ \\ & = 36.86^\circ, 323.14^\circ, 36.86^\circ + 360^\circ, 323.14^\circ - 360^\circ \phantom{000} \text{[Since } -40^\circ < 2x - 40^\circ < 680^\circ ] \\ & = 36.86^\circ, 323.14^\circ, 396.86^\circ, -36.86^\circ \\ \\ 2x & = 76.86^\circ, 363.14^\circ, 436.86^\circ, 3.14^\circ \\ \\ x & = 38.43^\circ, 181.57^\circ, 218.43^\circ, 1.57^\circ \\ & = 1.57^\circ, 38.43^\circ, 181.57^\circ, 218.43^\circ \\ & \approx 1.6^\circ, 38.4^\circ, 181.6^\circ, 218.4^\circ \end{align}

\begin{align} 10 \sin 2x & = 3 \\ \sin 2x & = {3 \over 10} \\ \sin 2x & = 0.3 \phantom{00000} [\text{1st & 2nd quadrants since } \sin 2x > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.3) \\ & = 0.3046 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{0} 0 < 2x < 4\pi \\ \\ 2x & = 0.3046, \pi - 0.3046 \\ & = 0.3046, 2.836 \\ & = 0.3046, 2.836, 0.3046 + 2\pi, 2.836 + 2\pi \phantom{000} [\text{Since } 0 < 2x < 4\pi] \\ & = 0.3046, 2.836, 6.5877, 9.1191 \\ \\ x & = 0.1523, 1.418, 3.29385, 4.55955 \\ & \approx 0.152, 1.42, 3.29, 4.56 \end{align}

\begin{align} \cot 2x & = -2 \\ {1 \over \tan 2x} & = -2 \\ 1 & = -2\tan 2x \\ -{1 \over 2} & = \tan 2x \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan 2x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 2\right) \\ & = 0.4636 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{0} 0 < 2x < 4\pi \\ \\ 2x & = \pi - 0.4636, 2\pi - 0.4636 \\ & = 2.6779, 5.8195 \\ & = 2.6779, 5.8195, 2.6779 + 2\pi, 5.8195 + 2\pi \phantom{00000} [\text{Since } 0 < 2x < 4\pi] \\ & = 2.6779, 5.8195, 8.9610, 12.1026 \\ \\ x & = 1.33895, 2.90975, 4.4805, 6.0513 \\ & \approx 1.34, 2.91, 4.48, 6.05 \end{align}

\begin{align} \tan \left({1 \over 2}x\right) & = {3 \over 2} \phantom{00000} \left[ \text{1st & 3rd quadrants since } \tan {1 \over 2}x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(3 \over 2\right) \\ & = 0.9827 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{0} 0 < {1 \over 2}x < \pi \\ \\ {1 \over 2} x & = 0.9827, \pi + 0.9827 \\ \\ & = 0.9827, 4.1242 \left( \text{Reject, since } 0 < {1 \over 2}x < \pi \right) \\ \\ & = 0.9827 \\ \\ x & = 0.9827*2 \\ & = 1.9654 \\ & \approx 1.97 \end{align}

\begin{align} 4\sec 2x + 5 & = 0 \\ 4\sec 2x & = -5 \\ \sec 2x & = -{5 \over 4} \\ {1 \over \cos 2x} & = -{5 \over 4} \\ 4(1) & = -5(\cos 2x) \\ -{4 \over 5} & = \cos 2x \phantom{00000} \left[ \text{2nd & 3rd quadrant since } \cos 2x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(4 \over 5\right) \\ & = 0.6435 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{0} 0 < 2x < 4\pi \\ \\ 2x & = \pi - 0.6435, \pi + 0.6435 \\ & = 2.4980, 3.7850 \\ & = 2.4980, 3.7850, 2.4980 + 2\pi, 3.7850 + 2\pi \phantom{00000} [\text{Since } 0 < 2x < 4\pi] \\ & = 2.4980, 3.7850, 8.7811, 10.0681 \\ \\ x & = 1.249, 1.8925, 4.39055, 5.03405 \\ & \approx 1.25, 1.89, 4.39, 5.03 \end{align}

\begin{align} R & = 64 \sin 2\theta \\ \\ \text{Since baseball was hit} & \text{ upwards at angle } \theta \text{ to the horizontal,} \\ \\ 0^\circ \le \theta \le 180^\circ & \rightarrow \phantom{.} 0^\circ \le 2\theta \le 360^\circ \\ \\ \\ \text{When } & R = 55, \\ 55 & = 64\sin 2\theta \\ {55 \over 64} & = \sin 2\theta \phantom{00000} \left[ \text{1st & 2nd quadrant since } \sin 2\theta > 0 \right]\\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(55 \over 64\right) \\ & = 59.24^\circ \end{align}

\begin{align} 2\theta & = 59.24^\circ, 180^\circ - 59.24^\circ \\ & = 59.24^\circ, 120.76^\circ \\ \\ \theta & = 29.62^\circ, 60.38^\circ \\ & \approx 29.6^\circ, 60.4^\circ \end{align}

\begin{align} 3 \sin x + 2\cos x & = 2(\sin x + 3\cos x) \\ 3\sin x + 2\cos x & = 2\sin x + 6\cos x \\ 3\sin x - 2\sin x & = 6\cos x - 2\cos x \\ \sin x & = 4\cos x \\ {\sin x \over \cos x} & = 4 \\ \\ \tan x & = 4 \phantom{00000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \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}

\begin{align} {\sin x - 3\cos x \over 2\sin x + \cos x} = 2 \\ \sin x - 3\cos x & = 2(2\sin x + \cos x) \\ \sin x - 3\cos x & = 4\sin x + 2\cos x \\ \sin x - 4\sin x & = 2\cos x + 3\cos x \\ -3\sin x & = 5\cos x \\ {-3 \sin x \over \cos x} & = 5 \\ {\sin x \over \cos x} & = -{5 \over 3} \\ \\ \tan x & = -{5 \over 3} \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(5 \over 3\right) \\ & = 59.03^\circ \end{align}

\begin{align} x & = 180^\circ - 59.03^\circ, 360^\circ - 59.03^\circ \\ & = 120.97^\circ, 300.97^\circ \\ & \approx 121.0^\circ, 301.0^\circ \end{align}

\begin{align} 2\sin x & = \tan x \\ 2\sin x & = {\sin x \over \cos x} \\ 2\sin x \cos x & = \sin x \\ 2\sin x \cos x - \sin x & = 0 \\ \sin x (2\cos x - 1) & = 0 \\ \\ \sin x = 0 \phantom{00}&\text{or}\phantom{00} 2\cos x - 1 = 0 \end{align}

$$ \sin x = 0 $$

$$ x = 0^\circ \text{ (Reject)}, 180^\circ, 360^\circ \text{ (Reject)} $$

\begin{align} 2\cos x - 1 & = 0 \\ 2\cos x & = 1 \\ \cos x & = {1 \over 2} \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 2\right) \\ & = 60^\circ \end{align}

\begin{align} x & = 60^\circ, 360^\circ - 60^\circ \\ & = 60^\circ, 300^\circ \\ \\ \\ \therefore x & = 60^\circ, 180^\circ, 300^\circ \end{align}

\begin{align} \sec^2 x & = 4\sec x - 3 \\ \sec^2 x - 4\sec x + 3 & = 0 \\ (\sec x - 1)(\sec x - 3) & = 0 \\ \\ \sec x - 1 = 0 \phantom{00}&\text{or}\phantom{00} \sec x - 3 = 0 \end{align}

\begin{align} \sec x - 1 & = 0 \\ \sec x & = 1 \\ {1 \over \cos x} & = 1 \\ 1 & = \cos x \end{align}

$$ x = 0^\circ \text{ (Reject)}, 360^\circ \text{ (Reject)} $$

\begin{align} \sec x - 3 & = 0 \\ \sec x & = 3 \\ {1 \over \cos x } & = 3 \\ 1 & = 3\cos x \\ {1 \over 3} & = \cos x \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 3\right) \\ & = 70.52^\circ \end{align}

\begin{align} x & = 70.52^\circ, 360^\circ - 70.52^\circ \\ & = 70.52^\circ, 289.48^\circ \\ & \approx 70.5^\circ, 289.5^\circ \end{align}

\begin{align} 2\sin^2 x + 5\cos x + 1 & = 0 \\ 2(1 - \cos^2 x) + 5\cos x + 1 & = 0 \phantom{00000} [\text{Since } \sin^2 A + \cos^2 A = 1 \phantom{0} \rightarrow \phantom{0} \sin^2 A = 1 - \cos^2 A] \\ 2 - 2\cos^2 x + 5\cos x + 1 & = 0 \\ -2\cos^2 x + 5\cos x + 3 & = 0 \\ 2\cos^2 x - 5\cos x - 3 & = 0 \\ (2\cos x + 1)(\cos x - 3) & = 0 \\ \\ 2\cos x + 1 = 0 \phantom{00-}&\text{or}\phantom{00} \cos x - 3 = 0 \\ &\phantom{or00-3} \cos x = 3 \text{ (Reject, since } \cos x \le 1) \\ \\ \\ 2\cos x + 1 & = 0 \\ 2\cos x & = - 1 \\ \cos x & = -{1 \over 2} \phantom{00000} \left[ \text{2nd & 3rd quadrants since } \cos x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 2\right) \\ & = 60^\circ \end{align}

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

\begin{align} 6 - 2\cos^2 x & = 9\sin x \\ 6 - 2(1 - \sin^2 x) & = 9\sin x \phantom{00000} [\text{Since } \sin^2 A + \cos^2 A = 1 \phantom{0} \rightarrow \phantom{0} \cos^2 A = 1 - \sin^2 A] \\ 6 - 2 + 2\sin^2 x & = 9 \sin x \\ 2\sin^2 x - 9\sin x + 4 & = 0 \\ (2\sin x - 1)(\sin x - 4) & = 0 \\ \\ 2\sin x - 1 = 0 \phantom{00}&\text{or}\phantom{00} \sin x - 4 = 0 \\ &\phantom{or00-4} \sin x = 4 \text{ (Reject, since } \sin x \le 1) \\ \\ \\ 2\sin x - 1 & = 0 \\ 2\sin x & = 1 \\ \sin x & = {1 \over 2} \phantom{00000} \left[ \text{1st & 2nd quadrants since } \sin x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 2\right) \\ & = 30^\circ \end{align}

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

\begin{align} 7\cos x - 3 & = {2 \over \sec^2 x} \\ 7\cos x - 3 & = {2 \over {1 \over \cos^2 x}} \\ 7\cos x - 3 & = 2\cos^2 x \\ 0 & = 2\cos^2 x - 7\cos x + 3 \\ 0 & = (2\cos x - 1)(\cos x - 3) \\ \\ 2\cos x - 1 = 0 \phantom{00}&\text{or}\phantom{00} \cos x - 3 = 0 \\ &\phantom{or00-3} \cos x = 3 \text{ (Reject, since } \cos x \le 1) \\ \\ \\ 2\cos x - 1 & = 0 \\ 2\cos x & = 1 \\ \cos x & = {1 \over 2} \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 2\right) \\ & = 60^\circ \end{align}

\begin{align} x & = 60^\circ, 360^\circ - 60^\circ \\ & = 60^\circ, 300^\circ \end{align}

\begin{align} 2\tan x & = 3 + 2\cot x \\ 2\tan x & = 3 + 2\left(1 \over \tan x\right) \\ 2\tan^2 x & = 3\tan x + 2 \\ 2\tan^2 x - 3\tan x - 2 & = 0 \\ (2\tan x + 1)(\tan x - 2) & = 0 \\ \\ 2\tan x + 1 = 0 \phantom{00}&\text{or}\phantom{00} \tan x - 2 = 0 \end{align}

\begin{align} 2\tan x + 1 & = 0 \\ 2\tan x & = -1 \\ \tan x & = -{1 \over 2} \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 2\right) \\ & = 26.56^\circ \end{align}

\begin{align} x & = 180^\circ - 26.56^\circ, 360^\circ -26.56^\circ \\ & = 153.44^\circ, 333.44^\circ \\ & \approx 153.4^\circ, 333.4^\circ \end{align}

\begin{align} \tan x - 2 & = 0 \\ \tan x & = 2 \phantom{00000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2) \\ & = 63.43^\circ \end{align}

\begin{align} x & = 63.43^\circ, 180^\circ + 63.43^\circ \\ & = 63.43^\circ, 243.43^\circ \\ & \approx 63.4^\circ, 243.4^\circ \\ \\ \\ \therefore x & = 63.4^\circ, 153.4^\circ, 243.4^\circ, 333.4^\circ \end{align}

\begin{align} 2\tan^2 x - 1 & = 5\sec x \\ 2(\sec^2 x - 1) - 1 & = 5\sec x \phantom{00000} [\text{Since } \tan^2 A + 1 = \sec^2 A \phantom{0} \rightarrow \phantom{0} \tan^2 A = \sec^2 A - 1] \\ 2\sec^2 x - 2 - 1 & = 5\sec x \\ 2\sec^2 x - 5\sec x - 3 & = 0 \\ (2\sec x + 1)(\sec x - 3) & = 0 \\ \\ 2\sec x + 1 = 0 \phantom{00}&\text{or}\phantom{00} \sec x - 3 = 0 \end{align}

\begin{align} 2\sec x + 1 & = 0 \\ 2\sec x & = -1 \\ \sec x & = -{1 \over 2} \\ {1 \over \cos x} & = -{1 \over 2} \\ 2 & = -\cos x \\ \\ \cos x & = - 2 \phantom{00} \text{ (Reject, since } \cos x \ge -1) \end{align}

\begin{align} \sec x - 3 & = 0 \\ \sec x & = 3 \\ {1 \over \cos x} & = 3 \\ 1 & = 3\cos x \\ \\ \cos x & = {1 \over 3} \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 3\right) \\ & = 1.2309 \end{align}

\begin{align} x & = 1.2309, 2\pi - 1.2309 \\ & = 1.2309, 5.0522 \\ & \approx 1.23, 5.05 \end{align}

\begin{align} \text{cosec}^2 x + 2\cot x & = 4 \\ (1 + \cot^2 x) + 2\cot x & = 4 \phantom{00000} [\text{Identity: } 1 + \cot^2 x = \text{cosec}^2 x] \\ 1 + \cot^2 x + 2\cot x - 4 & = 0 \\ \cot^2 x + 2\cot x - 3 & = 0 \\ (\cot x + 3)(\cot x - 1) & = 0 \\ \\ \cot x + 3 = 0 \phantom{00}&\text{or}\phantom{00} \cot x - 1 = 0 \end{align}

\begin{align} \cot x + 3 & = 0 \\ \cot x & = -3 \\ {1 \over \tan x} & = -3 \\ 1 & = -3\tan x \\\ \\ \tan x & = -{1 \over 3} \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 3\right) \\ & = 0.32175 \end{align}

\begin{align} x & = \pi- 0.32175, 2\pi - 0.32175 \\ & = 2.8198, 5.9614 \\ & \approx 2.82, 5.96 \end{align}

\begin{align} \cot x - 1 & = 0 \\ \cot x & = 1 \\ {1 \over \tan x} & = {1 \over 1} \\ \tan x & = 1 \phantom{00000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (1) \\ & = 45^\circ \\ & = {\pi \over 4} \end{align}

\begin{align} x & = {\pi \over 4}, \pi + {\pi \over 4} \\ & = {\pi \over 4}, {5\pi \over 4} \\ \\ \\ \therefore x & = {\pi \over 4}, 2.82, {5\pi \over 4}, 5.96 \end{align}

\begin{align} 2\cos^2 x + \sin 20^\circ & = 1 \\ 2\cos^2 x + 0.34202 & = 1 \\ 2\cos^2 x & = 1 - 0.34202 \\ 2\cos^2 x & = 0.65798 \\ \cos^2 x & = 0.32899 \\ \cos x & = \pm \sqrt{0.32899} \phantom{00000} [\text{All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \sqrt{0.32899} \\ & = 54.99^\circ \end{align}

\begin{align} x & = 54.99^\circ, 180^\circ - 54.99^\circ, 180^\circ + 54.99^\circ, 360^\circ - 54.99^\circ \\ & = 54.99^\circ, 125.01^\circ, 234.99^\circ \text{ (Reject)} , 305.01^\circ \text{(Reject)} \\ & \approx 55.0^\circ, 125.0^\circ \end{align}

\begin{align} |2\cos x + 3\sin x| & = \sin x \\ 2\cos x + 3\sin x & = \pm\sin x \\ \\ 2\cos x + 3\sin x = \sin x \phantom{00}&\text{or}\phantom{00} 2\cos x + 3\sin x = - \sin x \end{align}

\begin{align} 2\cos x + 3\sin x & = \sin x \\ 3\sin x - \sin x & = -2\cos x \\ 2\sin x & = -2\cos x \\ \sin x & = -\cos x \\ {\sin x \over \cos x} & = - 1 \\ \\ \tan x & = -1 \phantom{00000} [\text{2nd & 4th quadrants since } \tan x < 0 ] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (1) \\ & = 45^\circ \end{align}

\begin{align} x & = 180^\circ - 45^\circ, 360^\circ - 45^\circ \\ & = 135^\circ, 315^\circ \text{ (Reject, since } 0^\circ < x < 180^\circ) \\ & = 135^\circ \end{align}

\begin{align} 2\cos x + 3\sin x & = -\sin x \\ 3\sin x + \sin x & = - 2\cos x \\ 4\sin x & = -2 \cos x \\ \sin x & = -{2 \over 4} \cos x \\ {\sin x \over \cos x} & = -{1 \over 2} \\ \\ \tan x & = -{1 \over 2} \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 2\right) \\ & = 26.56^\circ \end{align}

\begin{align} x & = 180^\circ - 26.56^\circ, 360^\circ - 26.56^\circ \\ & = 153.44^\circ, 333.44^\circ \text{ (Reject, since } 0^\circ < x < 180^\circ) \\ & \approx 153.4^\circ \\ \\ \\ \therefore x & = 135^\circ, 153.4^\circ \end{align}

\begin{align} \cos (2x - 20^\circ) & = 0.95 \phantom{00000} [\text{1st & 4th quadrants since } \cos (2x - 20^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.95) \\ & = 18.19^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 720^\circ \rightarrow \phantom{.} -20^\circ < 2x - 20^\circ < 700^\circ \\ \\ 2x - 20^\circ & = 18.19^\circ, 360^\circ - 18.19^\circ \\ & = 18.19^\circ, 341.81^\circ \\ & = 18.19^\circ, 341.81^\circ, 18.19^\circ + 360^\circ, 341.81^\circ - 360^\circ \phantom{00} \text{ [Since } -20^\circ < 2x - 20^\circ < 700^\circ] \\ & = 18.19^\circ, 341.81^\circ, 378.19^\circ, -18.19^\circ \\ & = -18.19^\circ, 18.19^\circ, 341.81^\circ, 378.19^\circ \\ \\ 2x & = 1.81^\circ, 38.19^\circ, 361.81^\circ, 398.19^\circ \\ \\ x & = 0.905^\circ, 19.095^\circ, 180.905^\circ, 199.095^\circ \\ & \approx 0.9^\circ, 19.1^\circ, 180.9^\circ, 199.1^\circ \end{align}

\begin{align} \cot (2x + 10^\circ) & = -0.5 \\ {1 \over \tan (2x + 10^\circ)} & = -{1 \over 2} \\ \\ \tan (2x + 10^\circ) & = - 2 \phantom{00000} [\text{2nd & 4th quadrants since } \tan (2x + 10^\circ) < 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2) \\ & = 63.43^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 720^\circ \rightarrow \phantom{.} 10^\circ < 2x + 10^\circ < 730^\circ \\ \\ 2x + 10^\circ & = 180^\circ - 63.43^\circ, 360^\circ - 63.43^\circ \\ & = 116.57^\circ, 296.57^\circ \\ & = 116.57^\circ, 296.57^\circ, 116.57^\circ + 360^\circ, 296.57^\circ + 360^\circ \phantom{00} \text{ [Since } 10^\circ < 2x + 10^\circ < 730^\circ] \\ & = 116.57^\circ, 296.57^\circ, 476.57^\circ, 656.57^\circ \\ \\ 2x & = 106.57^\circ, 286.57^\circ, 466.57^\circ, 646.57^\circ \\ \\ x & = 53.285^\circ, 143.285^\circ, 233.285^\circ, 323.285^\circ \\ & \approx 53.3^\circ, 143.3^\circ, 233.3^\circ, 323.3^\circ \end{align}

\begin{align} \text{cosec } (2x + 60^\circ) & = 4 \\ {1 \over \sin (2x + 60^\circ)} & = 4 \\ 1 & = 4\sin (2x + 60^\circ) \\ \\ \sin (2x + 60^\circ) & = {1 \over 4} \phantom{00000} \left[ \text{1st & 2nd quadrants since } \sin (2x + 60^\circ) > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 4\right) \\ & = 14.47^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 720^\circ \rightarrow \phantom{.} 60^\circ < 2x + 60^\circ < 780^\circ \\ \\ 2x + 60^\circ & = 14.47^\circ, 180^\circ - 14.47^\circ \\ & = 14.47^\circ, 165.53^\circ \\ & = 14.47^\circ \text{ (Reject)}, 165.53^\circ, 14.47^\circ + 360^\circ, 165.53^\circ + 360^\circ, 14.47^\circ + 360^\circ + 360^\circ \\ & = 165.53^\circ, 374.47^\circ, 525.53^\circ, 734.47^\circ \\ \\ 2x & = 105.53^\circ, 314.47^\circ, 465.53^\circ, 674.47^\circ \\ \\ x & = 52.765^\circ, 157.235^\circ, 232.765^\circ, 337.235^\circ \\ & \approx 52.8^\circ, 157.2^\circ, 232.8^\circ, 337.2^\circ \end{align}

\begin{align} \tan (2x - 60^\circ) & = -1 \phantom{00000} [\text{2nd & 4th quadrants since } \tan (2x - 60^\circ) < 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (1) \\ & = 45^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 720^\circ \rightarrow \phantom{.} -60^\circ < 2x - 60^\circ < 660^\circ \\ \\ 2x - 60^\circ & = 180^\circ - 45^\circ, 360^\circ - 45^\circ \\ & = 135^\circ, 315^\circ \\ & = 135^\circ, 315^\circ, 135^\circ + 360^\circ, 315^\circ - 360^\circ \phantom{000} \text{ [Since } -60^\circ < 2x - 60^\circ < 660^\circ] \\ & = 135^\circ, 315^\circ, 495^\circ, -45^\circ \\ & = -45^\circ, 135^\circ, 315^\circ, 495^\circ \\ \\ 2x & = 15^\circ, 195^\circ, 375^\circ, 555^\circ \\ \\ x & = 7.5^\circ, 97.5^\circ, 187.5^\circ, 277.5^\circ \end{align}

\begin{align} \cot (3x + 0.5) & = 3 \\ {1 \over \tan (3x + 0.5)} & = 3 \\ 1 & = 3\tan (3x + 0.5) \\ \\ \tan (3x + 0.5) & = {1 \over 3} \phantom{00000} \left[ \text{1st & 3rd quadrants since } \tan (3x + 0.5) > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 3\right) \\ & = 0.3217 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{.} 0 < 3x < 6\pi \rightarrow \phantom{.} 0.5 < 3x + 0.5 < 6\pi + 0.5 \rightarrow \phantom{.} 0.5 < 3x + 0.5 < 19.3495 \\ \\ 3x + 0.5 & = 0.3217, \pi + 0.3217 \\ & = 0.3217, 3.4632 \\ & = 0.3217 \text{ (Reject)}, 3.4632, 0.3217 + 2\pi, 3.4632 + 2\pi, 0.3217 + 4\pi, 3.4632 + 4\pi, 0.3217 + 6\pi \\ & = 3.4632, 6.6048, 9.7463, 12.8880, 16.0295, 19.1712 \\ \\ 3x & = 2.9632, 6.1048, 9.2463, 12.3880, 15.5295, 18.6712 \\ \\ x & = 0.9877, 2.0349, 3.0821, 4.1293, 5.1765, 6.2237 \\ & \approx 0.988, 2.03, 3.08, 4.13, 5.18, 6.22 \end{align}

\begin{align} \sin \left({3x \over 2} + {5 \over 6}\right) & = {\sqrt{3} \over 4} \phantom{00000} \left[\text{1st & 2nd quadrant since } \sin \left({3x \over 2} + {5 \over 6}\right) > 0 \right]\\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(\sqrt{3} \over 4\right) \\ & = 0.4478 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{.} 0 < {3x \over 2} < 3\pi \rightarrow \phantom{.} {5 \over 6} < {3x \over 2} + {5 \over 6} < 3\pi + {5 \over 6} \rightarrow \phantom{.} 0.8333 < {3x \over 2} + {5 \over 6} < 10.2581 \\ \\ {3x \over 2} + {5 \over 6} & = 0.4478, \pi - 0.4478 \\ & = 0.4478, 2.6937 \\ & = 0.4478 \text{ (Reject)}, 2.6937, 0.4478 + 2\pi, 2.6937 + 2\pi \phantom{000} \left[ \text{Since } 0.8333 < {3x \over 2} + {5 \over 6} < 10.2581 \right] \\ & = 2.6937, 6.7309, 8.9768 \\ \\ {3x \over 2} & = 1.8603, 5.8975, 8.1434 \\ \\ x & = 1.2402, 3.9316, 5.4289 \\ & \approx 1.24, 3.93, 5.43 \end{align}

\begin{align} 3\text{cosec } (2x - 3) + 15 & = 1 \\ 3\text{cosec } (2x - 3) & = -14 \\ \text{cosec } (2x - 3) & = -{14 \over 3} \\ {1 \over \sin (2x - 3)} & = -{14 \over 3} \\ -3 & = 14\sin (2x - 3) \\ \\ \sin (2x - 3) & = -{3 \over 14} \phantom{00000} \left[ \text{3rd & 4th quadrants since } \sin (2x - 3) < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(3 \over 14\right) \\ & = 0.2159 \end{align}

\begin{align} \text{Since } 0 < x < 2\pi & \rightarrow \phantom{.} 0 < 2x < 4\pi \rightarrow \phantom{.} -3 < 2x - 3 < 4\pi - 3 \rightarrow \phantom{.} -3 < 2x - 3 < 9.5663 \\ \\ 2x - 3 & = \pi + 0.2159, 2\pi - 0.2159 \\ & = 3.3574, 6.0672 \\ & = 3.3574, 6.0672, 3.3574 - 2\pi, 6.0672 - 2\pi \phantom{00000} [\text{Since } -3 < 2x - 3 < 9.5663] \\ & = 3.3574, 6.0672, -2.9257, -0.2159 \\ & = -2.9257, -0.2159, 3.3574, 6.0672 \\ \\ 2x & = 0.0743, 2.7841, 6.3574, 9.0672 \\ \\ x & = 0.03715, 1.39205, 3.1787, 4.5336 \\ & \approx 0.0372, 1.39, 3.18, 4.53 \end{align}

\begin{align} \sin (3x + 70^\circ) & = 0.2 \phantom{00000} [\text{1st & 2nd quadrants since } \sin (3x + 70^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.2) \\ & = 11.53^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 180^\circ & \rightarrow \phantom{.} 0^\circ < 3x < 540^\circ \rightarrow \phantom{.} 70^\circ < 3x + 70^\circ < 610^\circ \\ \\ 3x + 70^\circ & = 11.53^\circ, 180^\circ - 11.53^\circ \\ & = 11.53^\circ, 168.47^\circ \\ & = 11.53^\circ \text{ (Reject)}, 168.47^\circ, 11.53^\circ + 360^\circ, 168.47^\circ + 360^\circ \phantom{00000} [\text{Since } 70^\circ < 3x + 70^\circ < 610^\circ] \\ & = 168.47^\circ, 371.53^\circ, 528.47^\circ \\ \\ 3x & = 98.47^\circ, 301.53^\circ, 458.47^\circ \\ \\ x & = 32.823^\circ, 100.51^\circ, 152.823^\circ \\ & \approx 32.8^\circ, 100.5^\circ, 152.8^\circ \end{align}

\begin{align} 8 \text{cosec } 2x \cot 2x & = 3 \\ 8 \left(1 \over \sin 2x\right) \left(\cos 2x \over \sin 2x\right) & = 3 \\ {8 \cos 2x \over \sin^2 2x} & = 3 \\ 8\cos 2x & = 3\sin^2 2x \\ 8\cos 2x & = 3(1 - \cos^2 2x) \phantom{000000} [\text{Since } \sin^2 A + \cos^2 A = 1 \phantom{0} \rightarrow \sin^2 A = 1 - \cos^2 A] \\ 8\cos 2x & = 3 - 3\cos^2 2x \\ 3\cos^2 2x + 8\cos 2x - 3 & = 0 \\ (3\cos 2x - 1)(\cos 2x + 3) & = 0 \\ \\ 3\cos 2x - 1 = 0 \phantom{00}&\text{or}\phantom{00} \cos 2x + 3 = 0 \\ 3\cos 2x = 1 \phantom{00}&\phantom{or00+ 3} \cos 2x = -3 \text{ (Reject, since } \cos 2x \le 1) \\ \\ \\ 3\cos 2x & = 1 \\ \cos 2x & = {1 \over 3} \phantom{00000} \left[ \text{1st & 4th quadrants since } \cos 2x > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 3\right) \\ & = 70.52^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 180^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 360^\circ \\ \\ 2x & = 70.52^\circ, 360^\circ - 70.52^\circ \\ & = 70.52^\circ, 289.48^\circ \\ \\ x & = 35.26^\circ, 144.74^\circ \\ & \approx 35.3^\circ, 144.7^\circ \end{align}

\begin{align} 2\sin x \cos x - 2 & = \cos x - 4\sin x \\ 2\sin x \cos x - \cos x + 4\sin x - 2 & = 0 \\ \\ (2\sin x - 1)(\cos x + 2) & = 0 \phantom{00000} \text{[Apply result from part (i)]} \\ \\ 2\sin x - 1 = 0 \phantom{00}&\text{or}\phantom{00} \cos x + 2 = 0 \\ 2\sin x = 1 \phantom{00}&\phantom{or00+2} \cos x = - 2 \text{ (Reject, since } \cos x \ge -1) \\ \\ \\ 2\sin x & = 1 \\ \sin x & = {1 \over 2} \phantom{00000} \left[\text{1st & 2nd quadrants since } \sin x > 0 \right]\\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 2\right) \\ & = 30^\circ \end{align}

\begin{align} x & = 30^\circ, 180^\circ - 30^\circ \\ & = 30^\circ, 150^\circ \\ & = 30^\circ, 150^\circ, 30^\circ - 360^\circ, 150^\circ - 360^\circ \phantom{000} [\text{Since } -360^\circ \le x \le 360^\circ] \\ & = 30^\circ, 150^\circ, -330^\circ, -210^\circ \\ & = -330^\circ, -210^\circ, 30^\circ, 150^\circ \end{align}

\begin{align} 2\tan^2 \theta & = 5\sec \theta + 10 \\ 2(\sec^2 \theta - 1) & = 5\sec \theta + 10 \phantom{000} [\text{Since } 1 + \tan^2 A = \sec^2 A \phantom{0} \rightarrow \tan^2 A = \sec^2 A - 1] \\ 2\sec^2 \theta - 2 & = 5\sec \theta + 10 \\ 2\sec^2 \theta - 5\sec \theta - 12 & = 0 \\ (2\sec \theta + 3)(\sec \theta - 4) & = 0 \end{align}

\begin{align} 2\sec \theta + 3 & = 0 &\text{or}\phantom{0000} \sec \theta - 4 & = 0 \\ 2\sec \theta & = -3 & \sec \theta & = 4 \\ \sec \theta & = -{3 \over 2} & \sec \theta & = 4 \\ {1 \over \cos \theta} & = -{3 \over 2} & {1 \over \cos \theta} & = 4 \\ 2 & = -3\cos \theta & 1 & = 4\cos \theta \\ -{2 \over 3} & = \cos \theta & {1 \over 4} & = \cos \theta \text{ (Reject)} \end{align}

(Since $\theta$ is obtuse, it lies in the 2nd quadrant. Thus $\cos \theta$ must be negative)

\begin{align} \cos \theta & = {Adj \over Hyp} \\ & = -{2 \over 3} \\ & = {-2 \over 3} \end{align}

\begin{align} \text{By Pytha} & \text{gora's theorem,} \\ 3^2 & = x^2 + (-2)^2 \\ 9 & = x^2 + 4 \\ 5 & = x^2 \\ \\ x & = \pm \sqrt{5} \\ & = \sqrt{5} \text{ or } -\sqrt{5} \text{ (Reject)} \\ \\ \tan \theta & = {Opp \over Adj} \\ & = {\sqrt{5} \over -2} \\ & = -{\sqrt{5} \over 2} \end{align}

\begin{align} 3\cos^2 2x + 4\sin 2x & = 1 \\ 3(1 - \sin^2 2x) + 4\sin 2x & = 1 \phantom{00000} [\text{Since } \sin^2 A + \cos^2 A = 1 \phantom{0} \rightarrow \cos^2 A = 1 - \sin^2 A] \\ 3 - 3\sin^2 2x + 4\sin 2x & = 1 \\ -3\sin^2 2x + 4\sin 2x + 2 & = 0 \\ 3\sin^2 2x - 4\sin 2x - 2 & = 0 \\ \\ \text{Let } & u = \sin 2x, \\ \\ 3u^2 - 4u - 2 & = 0 \\ \\ u & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-4) \pm \sqrt{(-4)^2 - 4(3)(-2)} \over 2(3)} \\ & = {4 \pm \sqrt{40} \over 6} \\ & = {4 \pm \sqrt{4}\sqrt{10} \over 6} \\ & = {4 \pm 2\sqrt{10} \over 6} \\ & = {2 \pm \sqrt{10} \over 3} \\ & = {2 + \sqrt{10} \over 3} \text{ or } {2 - \sqrt{10} \over 3} \\ \\ \\ \text{Since } & u = \sin 2x, \\ \\ \therefore \sin 2x & = {2 + \sqrt{10} \over 3} \text{ or } {2 - \sqrt{10} \over 3} \\ & = 1.7207 \text{ (Reject, since } \sin 2x \le 1) \text{ or } -0.3874 \\ \\ \\ \sin 2x & = -0.3874 \phantom{00000} [\text{3rd & 4th quadrants since } \sin 2x < 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.3874) \\ & = 22.79^\circ \end{align}

\begin{align} \text{Since } 0^\circ < x < 360^\circ & \rightarrow \phantom{.} 0^\circ < 2x < 720^\circ \\ \\ 2x & = 180^\circ + 22.79^\circ, 360^\circ - 22.79^\circ \\ & = 202.79^\circ, 337.21^\circ \\ & = 202.79^\circ, 337.21^\circ, 202.79^\circ + 360^\circ, 337.21^\circ + 360^\circ \phantom{00000} [\text{Since } 0^\circ < 2x < 720^\circ] \\ & = 202.79^\circ, 337.21^\circ, 562.79^\circ, 697.21^\circ \\ \\ x & = 101.395^\circ, 168.605^\circ, 281.395^\circ, 348.605^\circ \\ & \approx 101.4^\circ, 168.6^\circ, 281.4^\circ, 348.6^\circ \end{align}

\begin{align} \sec x (\tan x - 2) & = 2\text{cosec } x \\ {1 \over \cos x} (\tan x - 2) & = 2\left(1 \over \sin x\right) \\ {1 \over \cos x} (\tan x - 2) & = {2 \over \sin x} \\ {\sin x \over \cos x} (\tan x - 2) & = 2 \\ \tan x (\tan x - 2) & = 2 \\ \tan^2 x - 2\tan x - 2 & = 0 \\ \\ \text{Let } & u = \tan x, \\ \\ u^2 - 2u - 2 & = 0 \\ \\ u & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2)} \over 2(1)} \\ & = {2 \pm \sqrt{12} \over 2} \\ & = {2 \pm \sqrt{4}\sqrt{3} \over 2} \\ & = {2 \pm 2\sqrt{3} \over 2} \\ & = 1 \pm \sqrt{3} \\ \\ \tan x & = 1 \pm \sqrt{3} \\ & = 2.7320 \text{ or } -0.7320 \end{align}

\begin{align} \tan x & = 2.7320 \phantom{00000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2.7320) \\ & = 69.89^\circ \end{align}

\begin{align} x & = 69.89^\circ, 180^\circ + 69.89^\circ \\ & = 69.89^\circ, 249.89^\circ \\ & \approx 69.9^\circ, 249.9^\circ \end{align}

\begin{align} \tan x & = -0.7320 \phantom{00000} [\text{2nd & 4th quadrants since } \tan x < 0 ] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (0.7320) \\ & = 36.20^\circ \end{align}

\begin{align} x & = 180^\circ - 36.20^\circ, 360^\circ - 36.20^\circ \\ & = 143.8^\circ, 323.8^\circ \\ \\ \\ \therefore x & = 69.9^\circ, 143.8^\circ, 249.9^\circ, 323.8^\circ \end{align}

\begin{align} \sin 3x (4\sin 3x - 3\cos 3x) & = 4 \\ 4\sin^2 3x - 3\sin 3x \cos 3x & = 4 \\ 4\sin^2 3x - 3\sin 3x \cos 3x & = 4(1) \\ 4\sin^2 3x - 3\sin 3x \cos 3x & = 4(\sin^2 3x + \cos^2 3x) \phantom{00000} [\text{Identity: } \sin^2 A + \cos^2 A = 1] \\ 4\sin^2 3x - 3\sin 3x \cos 3x & = 4\sin^2 3x + 4\cos^2 3x \\ -3\sin 3x \cos 3x - 4\cos^2 3x & = 0 \\ 3\sin 3x \cos 3x + 4\cos^2 3x & = 0 \\ \cos 3x (3\sin 3x + 4\cos 3x) & = 0 \\ \\ \cos 3x = 0 \phantom{00}&\text{or}\phantom{00} 3\sin 3x + 4\cos 3x = 0 \\ \\ \\ \text{Since } 0 \le x \le \pi & \rightarrow \phantom{.} 0 \le 3x \le 3\pi \end{align}

$$ \cos 3x = 0 $$

\begin{align} 3x & = {\pi \over 2}, {3\pi \over 2} \\ & = {\pi \over 2}, {3\pi \over 2}, {\pi \over 2} + 2\pi \phantom{00000} [\text{Since } 0 \le 3x \le 3\pi] \\ & = {\pi \over 2}, {3\pi \over 2}, {5\pi \over 2} \\ \\ x & = {\pi \over 6}, {\pi \over 2}, {5\pi \over 6} \end{align}

\begin{align} 3\sin 3x + 4\cos 3x & = 0 \\ 3\sin 3x & = -4\cos 3x \\ {3\sin 3x \over \cos 3x} & = -4 \\ {\sin 3x \over \cos 3x} & = -{4 \over 3} \\ \\ \tan 3x & = -{4 \over 3} \phantom{00000} \left[ \text{2nd & 4th quadrants since } \tan 3x < 0 \right] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(4 \over 3\right) \\ & = 0.9272 \end{align}

\begin{align} 3x & = \pi - 0.9272, 2\pi - 0.9272 \\ & = 2.2143, 5.3559 \\ & = 2.2143, 5.3559, 2.2143 + 2\pi \phantom{00000} [\text{Since } 0 \le 3x \le 3\pi] \\ & = 2.2143, 5.3559, 8.4974 \\ \\ x & = 0.7381, 1.7853, 2.8324 \\ & \approx 0.738, 1.79, 2.83 \\ \\ \\ \therefore x & = {\pi \over 6}, {\pi \over 2}, {5\pi \over 6}, 0.738, 1.79, 2.83 \end{align}