\begin{align} 2\cos x - 1 & = 4 \cos x \\ 2\cos x - 4\cos x & = 1 \\ -2\cos x & = 1 \\ \cos x & = -{1 \over 2} \phantom{000000} [\text{2nd or 3rd quadrant}] \\ \\ \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} 4\sin^2 x & = 3 \\ \sin^2 x & = {3 \over 4} \\ \sin x & = \pm \sqrt{3 \over 4} \\ \sin x & = \pm {\sqrt{3} \over 2} \phantom{000000} [\text{All 4 quadrants}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(\sqrt{3} \over 2\right) \\ & = 60^\circ \\ & = {\pi \over 3} \end{align}

\begin{align} x & = {\pi \over 3}, \pi - {\pi \over 3}, \pi + {\pi \over 3}, 2\pi - {\pi \over 3} \\ & = {\pi \over 3}, {2\pi \over 3}, {4\pi \over 3}, {5\pi \over 3} \end{align}

$$ (2\sin x+ 1)(2\cos x + 5) = 0 $$ \begin{align} 2\sin x + 1 & = 0 & \text{ or } \phantom{000000} 2\cos x + 5 & = 0 \\ 2\sin x & = -1 & 2\cos x & = - 5 \\ \sin x & = {-1 \over 2} & \cos x & = -{5 \over 2} \\ & = -0.5 & & = -2.5 \text{ (Reject since } \cos x \ge -1 ) \end{align}
\begin{align} \sin x & = -0.5 \phantom{000000} [\text{3rd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.5) \\ & = 30^\circ \end{align}

\begin{align} x & = 180 + 30^\circ, 360^\circ - 30^\circ \\ & = 210^\circ, 330^\circ \end{align}

$$ (2\cos x - 1)(2\cot x + 3) = 0 $$ \begin{align} 2 \cos x - 1 & = 0 & \text{ or }\phantom{00000} 2\cot x + 3 & = 0 \end{align}
\begin{align} 2 \cos x - 1 & = 0 \\ 2 \cos x & = 1 \\ \cos x & = {1 \over 2} \phantom{000000} [\text{1st or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over 2\right) \\ & = 60^\circ \\ & = {\pi \over 3} \end{align}

\begin{align} x & = {\pi \over 3}, 2\pi - {\pi \over 3} \\ & = {\pi \over 3}, {5\pi \over 3} \\ \\ \\ 2 \cot x + 3 & = 0 \\ 2 \cot x & =- 3 \\ \cot x & = -{3 \over 2} \\ {1 \over \tan x} & = -{3 \over 2} \\ 2 & = -3 \tan x \\ -{2 \over 3} & = \tan x \phantom{000000} [\text{2nd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(2 \over 3\right) \\ & = 0.588 \end{align}

\begin{align} x & = \pi - 0.588, 2\pi - 0.588 \\ & = 2.5535, 5.6951 \\ & \approx 2.55, 5.70 \\ \\ \\ \therefore x & = {\pi \over 3}, {5\pi \over 3}, 2.55, 5.70 \end{align}

\begin{align} \tan A & > 0 \phantom{000000} [\text{1st or 3rd quadrant}] \\ \\ \cos B & < 0 \phantom{000000} [\text{2nd or 3rd quadrant}] \\ \\ \implies A & \text{ and } B \text{ are in the 3rd quadrant} \\ \\ \\ \tan A & = {3 \over 4} \\ {Opp \over Adj} & = {-3 \over -4} \end{align}

\begin{align} x & = \sqrt{(-3)^2 + (-4)^2} \\ & = 5 \\ \\ \sin A & = {Opp \over Hyp} \\ & = -{3 \over 5} \end{align}

\begin{align} \cos B & = -{3 \over 5} \\ {Adj \over Hyp} & = {-3 \over 5} \end{align}

\begin{align} \sqrt{ (5)^2 - (-3)^2} & = 4 \\ \\ x & = -4 \\ \\ \tan (-B) & = -\tan B \\ & = - {Opp \over Adj} \\ & = - {-4 \over -3} \\ & = -{4 \over 3} \end{align}

\begin{align} y & = a \cos bx + 7 \\ y & = 3 \cos 6x + 7 \\ \\ \text{Amplitude} & = 3 \\ \\ \text{Center line: } & y = 7 \\ \\ \text{Max. value} & = 7 + 3 = 10 \\ \text{Min. value} & = 7 - 3 = 4 \\ \\ \text{Period} & = 60^\circ \\ \\ \text{No. of cycles} & = {180^\circ \over 60^\circ} \\ & = 3 \end{align}

\begin{align} y & = 5 \tan 2x \\ \\ \text{Period} & = {\pi \over b} \\ & = {\pi \over 2} \\ \\ \text{No. of cycles} & = {2\pi \over {\pi \over 2}} \\ & = 4 \\ \\ \text{Vertical asymptotes: } & x = {\pi \over 4}, x = {3\pi \over 4}, x = {5\pi \over 4}, x = {7\pi \over 4} \end{align}

\begin{align} y & = {x \over \pi} \\ & = {1 \over \pi} x \phantom{000000} [y = mx + c] \\ \\ \text{When } & x = 0, \\ y & = {1 \over \pi}(0) \\ & = 0 \\ \\ \text{When } & x = \pi, \\ y & = {1 \over \pi} (\pi) \\ & = 1 \\ \\ \text{Line passes} & \text{ through } (0,0) \text{ and } (\pi, 1) \end{align}

\begin{align} x & = 5\pi \tan 2x \\ \underbrace{x \over \pi}_\text{Line} & = \underbrace{5\tan 2x}_\text{Curve} \\ \\ \therefore 3 & \text{ solutions [exclude (0,0)]} \end{align}

\begin{align} y & = 2 \sin x \\ \\ \text{Amplitude} & = 2 \\ \\ \text{Max. value} & = 2 \\ \text{Min. value} & = -2 \\ \\ \text{Period} & = {2\pi \over 1} \\ & = 2\pi \\ \\ \text{No. of cycles} & = {2\pi \over 2\pi} \\ & = 1 \\ \\ \\ y & = \cos 2x \\ \\ \text{Amplitude} & = 1 \\ \\ \text{Max. value} & = 1 \\ \text{Min. value} & = -1 \\ \\ \text{Period} & = {2\pi \over 2} \\ & = \pi \\ \\ \text{No. of cycles} & = {2\pi \over \pi} \\ & = 2 \end{align}

\begin{align} \cos 2x - 2\sin x & = 1 \\ \underbrace{\cos 2x}_\text{Second curve} & = \underbrace{2\sin x + 1}_\text{Third curve} \end{align}

Translate the graph of $y = 2 \sin x$ by 1 units in the positive $y$-direction:

$$ 4 \text{ solutions} $$

\begin{align} 2|\sec 2x| & = |\text{cosec } x | \\ 2 \left| 1 \over \cos 2x \right| & = \left| 1 \over \sin x \right| \\ 2 \left(1 \over |\cos 2x | \right) & = {1 \over |\sin x | } \\ { 2 \over |\cos 2x| } & = {1 \over |\sin x| } \\ 2|\sin x| & = |\cos 2x| \\ |2||\sin x| & = |\cos 2x| \\ |2\sin x| & = |\cos 2x| \end{align}

Obtain the modulus of the curves from (i) by reflecting the portions below the x-axis:

$$ 4 \text{ solutions} $$

\begin{align} (2\sin^2 x + \sin x)(\tan x - 3\cos 20^\circ) & = 0 \\ \sin x(2\sin x + 1)(\tan x - 3\cos 20^\circ) & = 0 \\ \\ \sin x = 0 \phantom{000} \text{ or } \phantom{000} 2 \sin x + 1 & = 0 \phantom{000} \text{ or } \phantom{000} \tan x - 3\cos 20^\circ = 0 \end{align}
$$ \sin x = 0 $$

\begin{align} x & = 0^\circ, 180^\circ, 360^\circ \\ \\ \\ 2\sin x + 1 & = 0 \\ 2\sin x & = - 1\\ \sin x & = {-1 \over 2} \\ \sin x & = -{1 \over 2} \phantom{000000} [\text{3rd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left(1 \over 2\right) \\ & = 30^\circ \end{align}

\begin{align} x & = 180^\circ + 30^\circ, 360^\circ - 30^\circ \\ & = 210^\circ, 330^\circ \\ \\ \\ \tan x - 3\cos 20^\circ & = 0 \\ \tan x & = 3\cos 20^\circ \\ \tan x & = 2.81907 \phantom{000000} [\text{1st or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (2.81907) \\ & = 70.469^\circ \end{align}

\begin{align} x & = 70.469^\circ, 180^\circ + 70.469^\circ \\ & = 70.469^\circ, 250.469^\circ \\ & \approx 70.5^\circ, 250.5^\circ \end{align}
$$ \therefore x = 0^\circ, 70.5^\circ, 180^\circ, 210^\circ, 250.5^\circ, 330^\circ, 360^\circ $$

\begin{align} |3 \tan x - 1| & = 2 \\ 3 \tan x - 1 & = \pm 2 \\ \\ \\ 3 \tan x - 1 & = 2 \\ 3 \tan x & = 3 \\ \tan x & = {3 \over 3} \\ \tan x & = 1 \phantom{000000} [\text{1st or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} (1) \\ & = 45^\circ \end{align}

\begin{align} x & = 45^\circ, 180^\circ + 45^\circ \\ & = 45^\circ, 225^\circ \\ \\ \\ 3 \tan x - 1 & = -2 \\ 3 \tan x & = - 1 \\ \tan x & = -{1 \over 3} \phantom{000000} [\text{2nd or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left(1 \over 3\right) \\ & = 18.434^\circ \end{align}

\begin{align} x & = 180^\circ - 18.434^\circ, 360^\circ - 18.434^\circ \\ & = 161.566^\circ, 341.566^\circ \\ & \approx 161.6^\circ, 341.6^\circ \\ \\ \\ \therefore x & = 45^\circ, 161.16^\circ, 225^\circ, 341.6^\circ \end{align}

\begin{align} \tan x (\sqrt{2}\cos x + 1) & = 0 \\ \\ \tan x = 0 \phantom{000} & \text{ or } \phantom{000} \sqrt{2}\cos x + 1 = 0 \end{align}
$$ \tan x = 0$$

\begin{align} x & = \pi \\ \\ \\ \sqrt{2}\cos x + 1 & = 0 \\ \sqrt{2}\cos x & = -1 \\ \cos x & = {-1 \over \sqrt{2}} \\ \cos x & = -{1 \over \sqrt{2}} \phantom{000000} [\text{2nd or 3rd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(1 \over \sqrt{2}\right) \\ & = {\pi \over 4} \end{align}

\begin{align} x & = \pi - {\pi \over 4}, \pi + {\pi \over 4} \\ & = {3\pi \over 4}, {5\pi \over 4} \\ \\ \therefore x & = {3\pi \over 4}, \pi, {5\pi \over 4} \end{align}

\begin{align} {2 \over \sec x - 1} & = 3 \\ 2 & = 3(\sec x - 1) \\ 2 & = 3\sec x - 3 \\ 5 & = 3\sec x \\ 5 & = 3 \left(1 \over \cos x\right) \\ 5 & = {3 \over \cos x} \\ 5\cos x & = 3 \\ \cos x & = {3 \over 5} \phantom{000000} [\text{1st or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(3 \over 5\right) \\ & = 0.92729 \end{align}

\begin{align} x & = 0.92729, 2\pi - 0.92729 \\ & = 0.92729, 5.3558 \\ & \approx 0.927, 5.36 \end{align}

\begin{align} \text{Since } -90^\circ < \tan^{-1} & \phantom{.} x < 90^\circ, \angle A \text{ is in the 4th quadrant} \\ \\ A & = \tan^{-1} (-2) \\ \tan A & = -2 \\ {Opp \over Adj} & = -{2 \over 1} \end{align}

\begin{align} x & = \sqrt{(-2)^2 + 1^2} \\ x & = \sqrt{5} \\ \\ \sin (-A) & = -\sin A \\ & = - \left(-{2 \over \sqrt{5}}\right) \\ & = {2 \over \sqrt{5}} \end{align}

\begin{align} \text{Consider: } & y = 2 \cos x - 1 \\ \\ \text{Amplitude} & = 2 \\ \\ \text{Center line: } & y = -1 \\ \\ \text{Max. value} & = -1 + 2 = 1 \\ \text{Min. value} & = -1 - 2 = -3 \\ \\ \text{Period} & = {2\pi \over 1} \\ & = 2\pi \\ \\ \text{No. of cycles} & = {2\pi \over 2\pi} \\ & = 1 \end{align}

Obtain the modulus graph by reflecting the portions below the x-axis:

\begin{align} y & = -{2x \over 3\pi} + 1 \\ y & = -{2 \over 3\pi} x + 1 \\ \\ \text{When } & x = 0, \\ y & = -{2 \over 3\pi} (0) + 1 \\ & = 0 + 1 \\ & = 1 \\ \\ \text{When } & x = \pi, \\ y & = -{2 \over 3\pi} (\pi) + 1 \\ & = -{2 \over 3} + 1 \\ & = {1 \over 3} \\ \\ \text{Line passes} & \text{ through } (0, 1) \text{ and } \left(\pi, {1 \over 3}\right) \end{align}

\begin{align} \sqrt{3} \tan x + 1 & = 0 \\ \sqrt{3} \tan x & = - 1 \\ \tan x & = {-1 \over \sqrt{3}} \\ \tan x & = -{1 \over \sqrt{3}} \phantom{000000} [\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} \text{Consider } & y = \sqrt{3} \tan x \\ \\ \text{Period} & = {180^\circ \over 1} \\ & = 180^\circ \\ \\ \text{No. of cycles} & = {360^\circ \over 180^\circ} \\ & = 2 \\ \\ \text{Vertical asymptotes: } & x = 90^\circ, x = 270^\circ \end{align}

Shift the graph up by 1 unit to obtain $y = \sqrt{3} \tan x + 1$:

\begin{align} y & = \cos 2x \\ \\ \text{Amplitude} & = 1 \\ \\ \text{Max. value} & = 1 \\ \text{Min. value} & = -1 \\ \\ \text{Period} & = {2\pi \over 2} \\ & = \pi \\ \\ \text{No. of cycles} & = {\pi \over \pi} \\ & = 1 \\ \\ \\ y & = 2 \sin 6x \\ \\ \text{Amplitude} & = 2 \\ \\ \text{Max. value} & = 2 \\ \text{Min. value} & = -2 \\ \\ \text{Period} & = {2\pi \over 6} \\ & = {\pi \over 3} \\ \\ \text{No. of cycles} & = {\pi \over {\pi \over 3}} \\ & = 3 \end{align}

\begin{align} \text{First value of } x & = \alpha \\ \\ \text{Other value of } x & = {\pi \over 2} + \alpha \end{align}

\begin{align} \text{Other value of } x & = \pi + \alpha \end{align}