(a)

\begin{align} \cos \theta + \sin \theta & = a \cos \theta + b \sin \theta \\ & = R \cos (\theta - \alpha) \\ \\ \therefore a = 1 &, b = 1 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + 1^2} \\ & = \sqrt{2} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1 \over 1} \right) \\ & = 45^\circ \\ \\ \\ \cos \theta + \sin \theta & = \sqrt{2} \cos (\theta - 45^\circ) \end{align}

(b)

\begin{align} \sqrt{3} \cos \theta - \sin \theta & = a \cos \theta - b \sin \theta \\ & = R \cos (\theta + \alpha) \\ \\ \therefore a = \sqrt{3} &, b = 1 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{(\sqrt{3})^2 + 1^2} \\ & = 2 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1 \over \sqrt{3}} \right) \\ & = 30^\circ \\ \\ \\ \sqrt{3} \cos \theta - \sin \theta & = 2\cos (\theta + 30^\circ) \end{align}

(c)

\begin{align} 3\sin \theta + 4\cos \theta & = a\sin \theta + b\cos \theta \\ & = R \sin (\theta + \alpha) \\ \\ \therefore a = 3 &, b = 4 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{3^2 + 4^2} \\ & = 5 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {4 \over 3} \right) \\ & = 53.1^\circ \\ \\ \\ 3\sin \theta + 4\cos \theta & = 5\sin (\theta + 53.1^\circ) \end{align}

(d)

\begin{align} \sin \theta - \sqrt{2} \cos \theta & = a\sin \theta - b\cos \theta \\ & = R\sin (\theta - \alpha) \\ \\ \therefore a = 1 &, b = \sqrt{2} \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + (\sqrt{2})^2} \\ & = \sqrt{3} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {\sqrt{2} \over 1} \right) \\ & = 54.7^\circ \\ \\ \\ \sin \theta - \sqrt{2} \cos \theta & = \sqrt{3} \sin (\theta - 54.7^\circ) \end{align}

\begin{align} 5 \sin \theta - 12 \cos \theta & = R \sin (\theta - \alpha) \\ & = a \sin \theta - b \cos \theta \\ \\ \therefore a = 5 &, b = 12 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{5^2 + 12^2} \\ & = 13 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {12 \over 5} \right) \\ & = 1.1760 \end{align}

(i)

\begin{align} I & = 15\sin (120\pi t) - 8 \cos (120 \pi t) \\ & = a\sin \theta - b \cos \theta \\ & = R\sin (\theta - \alpha) \\ \\ \therefore a = 15 & , b = 8 , \theta = 120\pi t \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{15^2 + 8^2} \\ & = 17 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {8 \over 15} \right) \\ & = 0.48996 \\ \\ \\ I & = 17 \sin (120\pi t - 0.48996) \end{align}

(ii)

\begin{align} \text{Since } -1 \le & \sin (120\pi t - 0.48996) \le 1, \\ -17 \le 17 &\sin (120 \pi t - 0.48996) \le 17 \\ \\ \therefore \text{Amplitude} & = 17\text{ A} \end{align}

(a)

\begin{align} y & = 3\sin \theta - 4\cos \theta \\ & = a\sin \theta - b\cos \theta \\ & = R \sin (\theta - \alpha) \\ \\ \therefore & \phantom{0} a = 3, b = 4 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{3^2 + 4^2} \\ & = 5 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left ( {4 \over 3} \right) \\ & = 53.13^\circ \\ \\ \\ y & = 5 \sin (\theta - 53.13^\circ) \\ \\ \text{Maximum} & = 5 \\ \\ \theta - 53.13^\circ & = 90^\circ \\ \theta & = 90^\circ + 53.13^\circ \\ & \approx 143.1^\circ \\ \\ \\ \text{Minimum} & = -5 \\ \\ \theta - 53.13^\circ & = 270^\circ \\ \theta & = 270^\circ + 53.13^\circ \\ & \approx 323.1^\circ \end{align}

(b)

\begin{align} y & = \cos \theta - 3\sin \theta \\ & = a\cos \theta - b \sin \theta \\ & = R \cos (\theta + \alpha) \\ \\ \therefore & \phantom{0} a = 1, b = 3 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + 3^2} \\ & = \sqrt{10} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {3 \over 1} \right) \\ & = 71.57^\circ \\ \\ \\ y & = \sqrt{10} \cos (\theta + 71.57^\circ) \\ \\ \text{Maximum} & = \sqrt{10} \end{align}
\begin{align} \theta + 71.57^\circ = 0^\circ \phantom{00000000000} & \text{or} \phantom{0000} \theta + 71.57^\circ = 360^\circ \\ \theta = - 71.57^\circ \text{ (N.A.)} \phantom{.} & \phantom{or0000+71.57^\circ} \theta = 360^\circ - 71.57^\circ \\ & \phantom{or0000+71.57^\circ} \theta \approx 288.4^\circ \end{align}
\begin{align} \text{Minimum } & = -\sqrt{10} \\ \\ \theta + 71.57^\circ & = 180^\circ \\ \theta & = 180^\circ - 71.57^\circ \\ & \approx 108.4^\circ \end{align}

(c)

\begin{align} y & = 6\cos \theta + 5\sin \theta \\ & = a\cos \theta + b\sin \theta \\ & = R \cos (\theta - \alpha) \\ \\ \therefore & \phantom{0} a = 6, b = 5 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{6^2 + 5^2} \\ & = \sqrt{61} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {5 \over 6} \right) \\ & = 39.81^\circ \\ \\ \\ y & = \sqrt{61} \cos (\theta - 39.81^\circ) \end{align}
\begin{align} \text{Maximum } & = \sqrt{61} \\ \\ \theta - 39.81^\circ = 0^\circ \phantom{000000} & \text{or} \phantom{000} \theta - 39.81^\circ = 360^\circ \\ \theta \approx 39.8^\circ \phantom{-.1^\circ} & \phantom{or000-39.71^\circ} \theta = 360^\circ + 39.81^\circ \\ & \phantom{or000-39.71^\circ} \theta = 399.81 \text{ (Reject)} \end{align}
\begin{align} \text{Minimum } & = -\sqrt{61} \\ \\ \theta -39.81^\circ & = 180^\circ \\ \theta & = 180^\circ + 39.81^\circ \\ & \approx 219.8^\circ \end{align}

(d)

\begin{align} y & = 3\sin \theta + 6\cos \theta \\ & = a \sin \theta + b\cos \theta \\ & = R\sin (\theta + \alpha) \\ \\ \therefore & \phantom{0} a = 3, b = 6 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{3^2 + 6^2} \\ & = \sqrt{45} \\ & = \sqrt{9} \sqrt{5} \\ & = 3\sqrt{5} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {6 \over 3} \right) \\ & = 63.43^\circ \\ \\ \\ y & = 3\sqrt{5} \sin (\theta + 63.43^\circ) \end{align}
\begin{align} \text{Maximum } & = 3\sqrt{5} \\ \\ \theta + 63.43^\circ & = 90^\circ \\ \theta & = 90^\circ - 63.43^\circ \\ \theta & \approx 26.6^\circ \end{align}
\begin{align} \text{Minimum } & = -3\sqrt{5} \\ \\ \theta + 63.43^\circ & = 270^\circ \\ \theta & = 270^\circ - 63.43^\circ \\ \theta & \approx 206.6^\circ \end{align}

(a)

\begin{align} 3\cos x - 4\sin x & = 1 \\ a\cos x - b\sin x & = 1 \\ R\cos (x + \alpha) & = 1 \\ \\ \therefore a = 3 &, b = 4 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{3^2 + 4^2} \\ & = 5 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {4 \over 3} \right) \\ & = 53.13^\circ \\ \\ 5\cos (x + 53.13^\circ) & = 1 \\ \cos (x + 53.13^\circ) & = 0.2 \phantom{000000} [\text{1st & 4th quadrants since } \cos (x + 53.13^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.2) \\ & = 78.46^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 53.13^\circ < \phantom{.} x & + 53.13^\circ < 413.13^\circ \\ \\ x + 53.13^\circ & = 78.46^\circ, 360^\circ - 78.46^\circ \\ & = 78.46^\circ, 281.54^\circ \\ \\ x & = 25.33^\circ, 228.41^\circ \\ & \approx 25.3^\circ, 228.4^\circ \end{align}

(b)

\begin{align} \sqrt{3} \sin x - \cos x & = 1 \\ a \sin x - b \cos x & = 1 \\ R \sin (x - \alpha) & = 1 \\ \\ \therefore a = \sqrt{3} &, b = 1 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{(\sqrt{3})^2 + 1^2} \\ & = 2 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1 \over \sqrt{3}} \right) \\ & = 30^\circ \\ \\ 2\sin (\theta - 30^\circ) & = 1 \\ \sin (\theta - 30^\circ) & = 0.5 \phantom{000000} [\text{1st & 2nd quadrants since } \sin (\theta - 30^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.5) \\ & = 30^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ -30^\circ < x & - 30^\circ < 330^\circ \\ \\ x - 30^\circ & = 30^\circ, 180^\circ - 30^\circ \\ & = 30^\circ, 150^\circ \\ \\ x & = 60^\circ, 180^\circ \end{align}

(c)

\begin{align} 6\cos x - 2\sin x & = 3.5 \\ a\cos x -b\sin x & = 3.5 \\ R\cos (x + \alpha) & = 3.5 \\ \\ \therefore a = 6 &, b = 2 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{6^2 + 2^2} \\ & = \sqrt{40} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {2 \over 6} \right) \\ & = 18.43^\circ \\ \\ \sqrt{40} \cos (x + 18.43^\circ) & = 3.5 \\ \cos (x + 18.43^\circ) & = {3.5 \over \sqrt{40}} \\ \cos (x + 18.43^\circ) & = 0.55340 \phantom{000000} [\text{1st & 4th quadrants since } \cos (x + 18.43^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.55340) \\ & = 56.4^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 18.43^\circ < x & + 18.43^\circ < 378.43^\circ \\ \\ x + 18.43^\circ & = 56.4^\circ, 360^\circ - 56.4^\circ \\ & = 56.4^\circ, 303.6^\circ \\ \\ x & = 37.97^\circ, 285.17^\circ \\ & \approx 38^\circ, 285.2^\circ \end{align}

(d)

\begin{align} \sin x + 2 \cos x & = \sqrt{2} \\ a\sin x + b \cos x & = \sqrt{2} \\ R\sin (x + \alpha) & = \sqrt{2} \\ \\ \therefore a = 1 &, b = 2 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + 2^2} \\ & = \sqrt{5} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {2 \over 1} \right) \\ & = 63.43^\circ \\ \\ \sqrt{5} \sin (x + 63.43^\circ) & = \sqrt{2} \\ \sin (x + 63.43^\circ) & = {\sqrt{2} \over \sqrt{5}} \\ \sin (x + 63.43^\circ) & = \sqrt{2 \over 5} \phantom{000000} [\text{1st & 2nd quadrants since } \sin (x + 63.43^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left( \sqrt{2 \over 5} \right) \\ & = 39.23^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 63.43^\circ < x & + 63.43^\circ < 423.43^\circ \\ \\ x + 63.43^\circ & = 39.23^\circ \text{ (N.A.)}, 180^\circ - 39.23^\circ, 39.23^\circ + 360^\circ \\ & = 140.77^\circ, 399.23^\circ \\ \\ x & = 77.34^\circ, 335.8^\circ \\ & \approx 77.3^\circ, 335.8^\circ \end{align}

(e)

\begin{align} 2.1 \cos x - \sin x & = 1.6 \\ a \cos x - b\sin x & = 1.6 \\ R \cos (x + \alpha) & = 1.6 \\ \\ \therefore a = 2.1 &, b = 1 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2.1^2 + 1^2} \\ & = \sqrt{5.41} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1 \over 2.1} \right) \\ & = 25.46^\circ \\ \\ \sqrt{5.41}\cos (x + 25.46^\circ) & = 1.6 \\ \cos (x + 25.46^\circ) & = {1.6 \over \sqrt{5.41}} \\ \cos (x + 25.46^\circ) & = 0.68789 \phantom{000000} [\text{1st & 4th quadrants since } \cos (x + 25.46^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.68789) \\ & = 46.54^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 25.46^\circ < x & + 25.46^\circ < 385.46^\circ \\ \\ x + 25.46^\circ & = 46.54^\circ, 360^\circ - 46.54^\circ \\ & = 46.54^\circ, 313.46^\circ \\ \\ x & = 21.08^\circ, 288^\circ \\ & \approx 21.1^\circ, 288^\circ \end{align}

(f)

\begin{align} \pi \cos x + e \sin x & = 2 \\ a \cos x + b \sin x & = 2 \\ R \cos (x - \alpha) & = 2 \\ \\ \therefore a = \pi &, b = e \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{\pi^2 + e^2} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {e \over \pi} \right) \\ & = 40.87^\circ \\ \\ \sqrt{\pi^2 + e^2} \cos (x - 40.87^\circ) & = 2 \\ \cos (x - 40.87^\circ) & = {2 \over \sqrt{\pi^2 + e^2}} \\ \cos (x - 40.87^\circ) & = 0.48142 \phantom{000000} [\text{1st & 4th quadrants since } \cos (x - 40.87^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.48142) \\ & = 61.22^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ -40.87^\circ < x & - 40.87^\circ < 319.13^\circ \\ \\ x - 40.87^\circ & = 61.22^\circ, 360^\circ - 61.22^\circ \\ & = 61.22^\circ, 298.78^\circ \\ \\ x & = 102.09^\circ, 339.65^\circ \\ & \approx 102.1^\circ, 339.7^\circ \end{align}

(i)

\begin{align} G(t) & = 8 \cos 4t + 6 \sin 4t \\ & = a \cos \theta + b \sin \theta \\ & = R \cos (\theta - \alpha) \\ \\ \therefore & \phantom{0} a = 8, b = 6, \theta = 4t \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{8^2 + 6^2} \\ & = 10 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {6 \over 8} \right) \\ & = 0.63450 \\ \\ \therefore G(t) & = 10 \cos (4t - 0.64350) \end{align}

(ii)

\begin{align} \text{Minimum} & = -10 \\ \\ 4t - 0.64350 & = \pi \\ 4t & = \pi + 0.64350 \\ & = 3.7851 \\ \\ t & = 0.946275 \\ & \approx 0.946s \end{align}

(i)

\begin{align} \text{When } & t = 0, \\ q & = 5e^{-10(0)}\{\cos [60(0)] + {1 \over 4} \sin [60(0)]\} \\ & = 5(1)\left[(1) + {1 \over 4}(0)\right] \\ & = 5 \\ \\ \therefore \text{Initial charge} & = 5 \text{ coulombs} \end{align}

(ii)

\begin{align} \cos 60t + {1 \over 4} \sin 60t & = a \cos \theta + b \sin \theta \\ & = R \cos (\theta - \alpha) \\ \\ \therefore a = 1 &, b = {1 \over 4}, \theta = 60t \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + \left( {1 \over 4} \right)^2} \\ & = \sqrt{17 \over 16} \\ & = {\sqrt{17} \over 4} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {{1 \over 4} \over 1} \right) \\ & = 0.24498 \\ \\ \\ \therefore q & = 5e^{-10t}\left( \cos 60t + {1 \over 4} \sin 60t \right) \\ & = 5e^{-10t} \left[ {\sqrt{17} \over 4}\cos (60t - 0.24498) \right] \\ & = {5\sqrt{17} \over 4} e^{-10t} \cos (60t - 0.24498) \\ \\ \therefore & \phantom{0} A = {5\sqrt{17} \over 4}, B = 0.245 \end{align}

(iii)

\begin{align} q & = {5\sqrt{17} \over 4} e^{-10t} \cos (60t - 0.24498) \\ & = {5\sqrt{17} \over 4} \left( {1 \over e^{10t}} \right) \cos (60t - 0.24498) \\ \\ \text{As } t & \rightarrow \infty, {1 \over e^{10t}} \rightarrow 0 \\ \\ q & = {5\sqrt{17} \over 4} (0) \cos (60t - 0.24498) \\ & = 0 \\ \\ \therefore \text{Value} & \text{ of charge} = 0 \text{ couloumb} \end{align}

(a)

\begin{align} 4\sin \theta + 3\cos \theta & = a \sin \theta + b\cos \theta \\ & = R \sin (\theta + \alpha) \\ \\ \therefore a = 4 &, b = 3 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{4^2 + 3^2} \\ & = 5 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {3 \over 4} \right) \\ & = 36.87^\circ \\ \\ \\ \therefore y & = 4\sin \theta + 3\cos \theta - 2 \\ & = 5\sin (\theta + 36.87^\circ) - 2 \end{align}
\begin{align} \text{Maximum} & = 5(1) - 2 \\ & = 3 \\ \\ \theta + 36.87^\circ & = 90^\circ \\ \theta & = 90^\circ - 36.87^\circ \\ & = 53.13^\circ \\ & \approx 53.1^\circ \end{align}
\begin{align} \text{Minimum}& = 5(-1) - 2 \\ & = -7 \\ \\ \theta + 36.87^\circ & = 270^\circ \\ \theta & = 270^\circ - 36.87^\circ \\ & = 233.13^\circ \\ & \approx 233.1^\circ \end{align}

(b)

\begin{align} 24\sin \theta - 7\cos \theta & = a\sin \theta - b\cos \theta \\ & = R\sin (\theta - \alpha) \\ \\ \therefore \phantom{0} & a = 24, b = 7 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{24^2 + 7^2} \\ & = 25 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {7 \over 24} \right) \\ & = 16.26^\circ \\ \\ \\ \therefore y & = 3 - 7\cos \theta + 24\sin \theta \\ & = 24\sin \theta - 7\cos \theta + 3 \\ & = 25 \sin (\theta - 16.26^\circ) + 3 \end{align}
\begin{align} \text{Maximum} & = 25(1) + 3 \\ & = 28 \\ \\ \theta - 16.26^\circ & = 90^\circ \\ \theta & = 90^\circ + 16.26^\circ \\ & = 106.26^\circ \\ & \approx 106.3^\circ \end{align}
\begin{align} \text{Minimum} & = 25(-1) + 3 \\ & = -22 \\ \\ \theta - 16.26^\circ & = 270^\circ \\ \theta & = 270^\circ + 16.26^\circ \\ & = 286.26^\circ \\ & \approx 286.3^\circ \end{align}

(c)

\begin{align} \sin \theta - \sqrt{2} \cos \theta & = a\sin \theta - b\cos \theta \\ & = R\sin (\theta - \alpha) \\ \\ \therefore a = 1 &, b = \sqrt{2} \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + (\sqrt{2})^2} \\ & = \sqrt{3} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {\sqrt{2} \over 1} \right) \\ & = 54.74^\circ \\ \\ \\ \therefore y & = \sin \theta - \sqrt{2} \cos \theta + 4\sqrt{3} \\ & = \sqrt{3} \sin (\theta - 54.74^\circ) + 4\sqrt{3} \end{align}
\begin{align} \text{Maximum} & = \sqrt{3}(1) + 4\sqrt{3} \\ & = 5\sqrt{3} \\ \\ \theta - 54.74^\circ & = 90^\circ \\ \theta & = 90^\circ + 54.74^\circ \\ & = 144.74^\circ \\ & \approx 144.7^\circ \end{align}
\begin{align} \text{Minimum} & = \sqrt{3}(-1) + 4\sqrt{3} \\ & = 3\sqrt{3} \\ \\ \theta - 54.74^\circ & = 270^\circ \\ \theta & = 270^\circ + 54.74^\circ \\ & = 324.74^\circ \\ & \approx 324.7^\circ \end{align}

(d)

\begin{align} 15\cos 2\theta + 8\sin 2\theta & = a\cos 2\theta + b\sin 2\theta \\ & = R \cos (2\theta - \alpha) \\ \\ \therefore & \phantom{0} a = 15, b = 8 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{15^2 + 8^2} \\ & = 17 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {8 \over 15} \right) \\ & = 28.07^\circ \\ \\ \\ \therefore y & = 7 + 15\cos 2\theta + 8\sin 2\theta \\ & = 7 + 17\cos (2\theta - 28.07^\circ) \end{align}
\begin{align} 0^\circ < \phantom{.} & \theta < 360^\circ \\ 0^\circ < \phantom{.} & 2\theta < 720^\circ \\ -28.07^\circ < 2\theta & - 28.07^\circ < 691.93^\circ \end{align} \begin{align} \text{Maximum} & = 7 + 17(1) \\ & = 24 \end{align} \begin{align} 2\theta - 28.07^\circ & = 0^\circ & \text{ or } \phantom{000} 2\theta - 28.07^\circ & = 360^\circ \\ 2\theta & = 28.07^\circ & 2\theta & = 388.07^\circ \\ \theta & = 14.035^\circ & \theta & = 194.035^\circ \\ \theta & \approx 14.0^\circ & \theta & \approx 194.0^\circ \end{align}
\begin{align} \text{Minimum} & = 7 + 17(-1) \\ & = -10 \end{align} \begin{align} 2\theta - 28.07^\circ & = 180^\circ & \text{or } \phantom{000} 2\theta - 28.07^\circ & = 540^\circ \\ 2\theta & = 208.07^\circ & 2\theta & = 568.07^\circ \\ \theta & = 104.035^\circ & \theta & = 284.035^\circ \\ \theta & \approx 104.0^\circ & \theta & \approx 284.0^\circ \end{align}

(i)

\begin{align} 0.5\sin \left( {\pi \over 6}x \right) - 1.2 \cos \left( {\pi \over 6}x\right) & = a\sin \theta - b\cos \theta \\ & = R\sin (\theta - \alpha) \\ \\ \therefore a = 0.5, b = 1.2 &, \theta = {\pi \over 6}x \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{0.5^2 + 1.2^2} \\ & = 1.3 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1.2 \over 0.5} \right) \\ & = 1.17601 \\ \\ \\ \therefore h & = 1.6 + 0.5\sin \left( {\pi \over 6}x \right) - 1.2 \cos \left( {\pi \over 6}x \right) \\ & = 1.6 + 1.3 \sin \left( {\pi \over 6}x - 1.17601 \right) \end{align}
\begin{align} \text{Maximum height} & = 1.6 + 1.3(1) \\ & = 2.9m \end{align} \begin{align} \text{Minimum height} & = 1.6 + 1.3(-1) \\ & = 0.3m \end{align}

(ii)

\begin{align} \text{When } & h = 2, \\ 2 & = 1.6 + 1.3 \sin \left( {\pi \over 6}x - 1.17601 \right) \\ 0.4 & = 1.3 \sin \left( {\pi \over 6}x - 1.17601 \right) \\ {0.4 \over 1.3} & = \sin \left( {\pi \over 6}x - 1.17601 \right) \\ \\ \sin \left( {\pi \over 6}x - 1.17601 \right) & = {4 \over 13} \phantom{000000} \left[ \text{1st & 2nd quadrants since } \sin \left( {\pi \over 6}x - 1.17601 \right) > 0 \right] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left( {4 \over 13} \right) \\ & = 0.31277 \end{align}

Note x is the number of hours after midnight and there are 24 hours in a day!

\begin{align} 0 < \phantom{.} & x < 24 \\ 0 < \phantom{.} & {1 \over 6}x < 4 \\ 0 < \phantom{.} & {\pi \over 6}x < 4\pi \\ 0 < {\pi \over 6}x & - 1.17601 < 11.39036 \end{align}
\begin{align} {\pi \over 6}x - 1.17601 & = 0.31277, \pi - 0.31277 \\ & = 0.31277, 2.82882, 0.31277 + 2\pi, 2.82882 + 2\pi \\ & = 0.31277, 2.82882, 6.59596, 9.11201 \\ \\ {\pi \over 6}x & = 1.48878, 4.00483, 7.77197, 10.28802 \\ \\ x & = 2.84336, 7.64866, 14.84337, 19.64867 \end{align}

\begin{align} 2.84336 \text{ hours} & = 2 \text{ hours} + (0.84336 \times 60) \text{ mins} \\ & = 2 \text{ hours} + 51 \text{ mins} \\ \\ 7.64866 \text{ hours} & = 7 \text{ hours} + (0.64866 \times 60) \text{ mins} \\ & = 7 \text{ hours} + 39 \text{ mins} \\ \\ 14.84337 \text{ hours} & = 14 \text{ hours} + (0.84337 \times 60) \text{ mins} \\ & = 14 \text{ hours} + 51 \text{ mins} \\ \\ 19.64867 \text{ hours} & = 19 \text{ hours} + (0.64867 \times 60) \text{ mins} \\ & = 19 \text{ hours} + 39 \text{ mins} \\ \end{align}

$$ \therefore \text{The times are } 02:51, 07:39, 14:51, 19:39 $$

(a)

\begin{align} {4\sin x - \cos x - 3 \over \sin x + \cos x} & = 2 \\ 4\sin x - \cos x - 3 & = 2(\sin x + \cos x) \\ 4\sin x - \cos x - 3 & = 2\sin x + 2\cos x \\ 4\sin x - 2\sin x - \cos x - 2\cos x & = 3 \\ 2\sin x - 3\cos x & = 3 \\ a\sin x + b\cos x & = 3 \\ R\sin (x - \alpha) & = 3 \\ \\ \therefore a = 2 &, b = 3 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2^2 + 3^2} \\ & = \sqrt{13} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {3 \over 2} \right) \\ & = 56.31^\circ \\ \\ \\ \therefore \sqrt{13} \sin (x - 56.31^\circ) & = 3 \\ \sin (x - 56.31^\circ) & = {3 \over \sqrt{13}} \phantom{000000} [\text{1st & 2nd quadrants since } \sin (x - 56.31^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left({3 \over \sqrt{13}}\right) \\ & = 56.31^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ -56.31^\circ < x & - 56.31^\circ < 304.69^\circ \\ \\ x - 56.31^\circ & = 56.31^\circ, 180^\circ - 56.31^\circ \\ & = 56.31^\circ, 123.69^\circ \\ \\ x & = 112.62^\circ, 180^\circ \\ & \approx 112.6^\circ, 180^\circ \end{align}

(b)

\begin{align} 2\cos 2x + \sin 2x + 2 & = 0 \\ 2\cos 2x + \sin 2x & = -2 \\ a\cos \theta + \sin \theta & = -2 \\ R\cos (\theta - \alpha) & = -2 \\ \\ \therefore a = 2, b = 1 &, \theta = 2x \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2^2 + 1^2} \\ & = \sqrt{5} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1 \over 2} \right) \\ & = 26.57^\circ \\ \\ \\ \therefore \sqrt{5} \cos (2x - 26.57^\circ) & = -2 \\ \cos (2x - 26.57^\circ) & = -{2 \over \sqrt{5}} \phantom{000000} [\text{2nd & 3rd quadrants since } \cos (2x - 26.57^\circ) < 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left( {2 \over \sqrt{5}} \right) \\ & = 26.57^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 0^\circ < \phantom{.} & 2x < 720^\circ \\ -26.57^\circ < 2x & - 26.57^\circ < 693.43^\circ \\ \\ 2x - 26.57^\circ & = 180^\circ - 26.57^\circ, 180^\circ + 26.57^\circ \\ & = 153.43^\circ, 206.57^\circ, 153.43^\circ + 360^\circ, 206.57^\circ + 360^\circ \\ & = 153.43^\circ, 206.57^\circ, 513.43^\circ, 566.57^\circ \\ \\ 2x & = 180^\circ, 233.14^\circ, 540^\circ, 593.14^\circ \\ \\ x & = 90^\circ, 116.57^\circ, 270^\circ, 296.57^\circ \\ & \approx 90^\circ, 116.6^\circ, 270^\circ, 296.6^\circ \end{align}

(c)

\begin{align} \sin 2x & = 2\sin x \cos x \\ \\ \text{Let } & x = {1 \over 2}x, \\ \sin \left[2\left( {1 \over 2}x \right)\right] & = 2\sin {1 \over 2}x \cos {1 \over 2}x \\ \sin x & = 2\sin {1 \over 2}x \cos {1 \over 2}x \\ \\ \\ \sin {1 \over 2} x \cos {1 \over 2}x + 2\cos x & = 1 \\ 2\sin {1 \over 2} x \cos {1 \over 2}x + 4\cos x & = 2 \\ \sin x + 4\cos x & = 2 \\ a\sin x + b\cos x & = 2 \\ R\sin (x + \alpha) & = 2 \\ \\ \therefore a = 1 &, b = 4 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{1^2 + 4^2} \\ & = \sqrt{17} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {4 \over 1} \right) \\ & = 75.96^\circ \\ \\ \\ \therefore \sqrt{17} \sin (x + 75.96^\circ) & = 2 \\ \sin (x + 75.96^\circ) & = {2 \over \sqrt{17}} \phantom{000000} [\text{1st & 2nd quadrants since } \sin (x + 75.96^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} \left({2 \over \sqrt{17}}\right) \\ & = 29.02^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 75.96^\circ < x & + 75.96^\circ < 435.96^\circ \\ \\ x + 75.96^\circ & = 29.02^\circ \text{ (N.A.)}, 180^\circ - 29.02^\circ \\ & = 150.98^\circ, 29.02^\circ + 360^\circ \\ & = 150.98^\circ, 389.02^\circ \\ \\ x & = 75.02^\circ, 313.06^\circ \\ & \approx 75.0^\circ, 313.1^\circ \end{align}

(d)

\begin{align} 2\cot x & = 3 + 2\text{ cosec }x \\ 2\left( {\cos x \over \sin x} \right) & = 3 + 2 \left({1 \over \sin x} \right) \\ {2\cos x \over \sin x} & = 3 + {2 \over \sin x} \\ 2\cos x & = \sin x \left(3 + {2 \over \sin x} \right) \\ 2\cos x & = 3\sin x + 2 \\ 2\cos x - 3\sin x & = 2 \\ a\cos x - b\sin x & = 2 \\ R\cos (x + \alpha) & = 2 \\ \\ \therefore a = 2 &, b = 3 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2^2 + 3^2} \\ & = \sqrt{13} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {3 \over 2} \right) \\ & = 56.31^\circ \\ \\ \\ \therefore \sqrt{13} \cos (x + 56.31^\circ) & = 2 \\ \cos (x + 56.31^\circ) & = {2 \over \sqrt{13}} \phantom{000000} [\text{1st & 4th quadrants since } \cos (x + 56.31^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left( {2 \over \sqrt{13}} \right) \\ & = 56.31^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 56.31^\circ < x & + 56.31^\circ < 416.31^\circ \\ \\ x + 56.31^\circ & = 56.31^\circ \text{ (N.A.)}, 360^\circ - 56.31^\circ \\ & = 303.69^\circ, 56.31^\circ + 360^\circ \\ & = 303.69^\circ, 416.31^\circ \text{ (N.A.)} \\ & = 303.69^\circ \\ \\ x & = 247.38^\circ \\ & \approx 247.4^\circ \end{align}

(e)

\begin{align} 2\sin x (2\sin x - \cos x) & = 1 \\ 4\sin^2 x - 2\sin x \cos x & = 1 \\ [\text{Double angle formula}] \phantom{000000} 4\sin^2 x - \sin 2x & = 1 \\ \\ \text{Since } \cos 2x & = 1 - 2\sin^2 x, \\ 2\sin^2 x & = 1 - \cos 2x \\ 4\sin^2 x & = 2 - 2\cos 2x \\ \\ \\ \therefore (2 - 2\cos 2x) - \sin 2x & = 1 \\ 2 - 2\cos 2x - \sin 2x & = 1 \\ -2\cos 2x - \sin 2x & = 1 - 2 \\ -2\cos 2x - \sin 2x & = -1 \\ 2\cos 2x + \sin 2x & = 1 \\ a\cos \theta + b\sin \theta & = 1 \\ R\cos (\theta - \alpha) & = 1 \\ \\ \therefore a = 2, b = 1 &, \theta = 2x \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2^2 + 1^1} \\ & = \sqrt{5} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {1 \over 2} \right) \\ & = 26.57^\circ \\ \\ \\ \therefore \sqrt{5} \cos (2x - 26.57^\circ) & = 1 \\ \cos (2x - 26.57^\circ) & = {1 \over \sqrt{5}} \phantom{000000} [\text{1st & 4th quadrants since } \cos (2x - 26.57^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left( {1 \over \sqrt{5}} \right) \\ & = 63.43^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & x < 360^\circ \\ 0^\circ < \phantom{.} & 2x < 720^\circ \\ -26.57^\circ < 2x & - 26.57^\circ < 693.43^\circ \\ \\ 2x - 26.57^\circ & = 63.43^\circ, 360^\circ - 63.43^\circ \\ & = 63.43^\circ, 296.57^\circ, 63.43^\circ + 360^\circ, 296.57^\circ + 360^\circ \\ & = 63.43^\circ, 296.57^\circ, 423.43^\circ, 656.57^\circ \\ \\ 2x & = 90^\circ, 323.14^\circ, 450^\circ, 683.14^\circ \\ \\ x & = 45^\circ, 161.57^\circ, 225^\circ, 341.57^\circ \\ & \approx 45^\circ, 161.6^\circ, 225^\circ, 341.6^\circ \end{align}

(i)

\begin{align} V & = 70\sin t + 85\cos t \\ & = a\sin t + b\cos t \\ & = R\sin (t + \alpha) \\ \\ \therefore & \phantom{0} a = 70, b = 85 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{70^2 + 85^2} \\ & = 110.11 \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {85 \over 70} \right) \\ & = 0.88187 \\ \\ \\ \therefore V & = 110.11 \sin (t + 0.88187) \end{align}

(ii)

\begin{align} \text{Maximum, } V & = 110.11(1) \\ & = 110 V \end{align} \begin{align} t + 0.88187 & = {\pi \over 2} \\ \\ t & = 0.68892 \\ & \approx 0.689 \text{ ms} \end{align}

(iii)

\begin{align} \text{When } & V = 50, \\ 50 & = 110.11 \sin (t + 0.88187) \\ \\ \sin (t + 0.88187) & = {50 \over 110.11} \\ \sin (t + 0.88187) & = 0.45409 \phantom{000000} [\text{1st & 2nd quadrants since } \sin (t + 0.88187) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.45409) \\ & = 0.47135 \end{align}

\begin{align} t + 0.88187 & = 0.47135, \pi - 0.47135 \\ & = 0.47135, 2.67024 \\ \\ t & = -0.41052 \text{ (N.A.)}, 1.78837 \\ & \approx 1.79 \\ \\ \therefore \text{First instant} & = 1.79 \text{ ms} \end{align}

(i)

\begin{align} x & = 10\cos 8t + 7\sin 8t \\ & = 10\cos \theta + b\sin \theta \\ & = R \cos (\theta - \alpha) \\ \\ \therefore & \phantom{0} a = 10, b = 7, \theta = 8t \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{10^2 + 7^2} \\ & = \sqrt{149} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {7 \over 10} \right) \\ & = 0.61073 \\ \\ \\ x & = \sqrt{149} \cos (8t - 0.61073) \end{align}

(ii)

\begin{align} \text{Maximum, } x & = \sqrt{149} (1) \\ & = \sqrt{149}\phantom{0}mm \\ \\ 8t - 0.61073 & = 0 \\ 8t & = 0.61073 \\ \\ t & = 0.076341 \\ \\ \therefore \text{First instant} & = 0.0763 \text{ s} \end{align}

(iii)

\begin{align} \text{When } & x = 9, \\ 9 & = \sqrt{149}\cos (8t - 0.61073) \\ {9 \over \sqrt{149}} & = \cos (8t - 0.61073) \\ \\ \therefore \cos (8t - 0.61073) & = 0.73731 \phantom{000000} [\text{1st & 4th quadrants since } \cos (8t - 0.61073) > 0] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} (0.73731) \\ & = 0.74172 \end{align}

\begin{align} 0 < \phantom{.} & t < 1 \\ 0 < \phantom{.} & 8t < 8 \\ -0.61073 < 8t & - 0.61073 < 7.30927 \\ \\ 8t - 0.61073 & = 0.74172, 2\pi - 0.74172, 0.74172 + 2\pi, (2\pi - 0.74172) + 2\pi \\ & = 0.74172, 5.54147, 7.02491, 11.82466 \text{ (N.A.)} \\ \\ 8t & = 1.35245, 6.1522, 7.63564 \\ \\ t & = 0.16905, 0.76902, 0.95445 \\ & \approx 0.169, 0.769, 0.954 \end{align}

Show question

\begin{align} \cos \angle QRS & = {Adj \over Hyp} \\ \cos \theta & = {SR \over QR} \\ \cos \theta & = {SR \over 8} \\ \\ SR & = 8\cos \theta \\ \\ \angle RQS & = 180^\circ - 90^\circ - \theta \\ & = 90^\circ - \theta \\ \\ \angle PQT & = 90^\circ - \angle RQS \\ & = 90^\circ - (90^\circ - \theta) \\ & = \theta \\ \\ \sin \angle PQT & = {Opp \over Hyp} \\ \sin \theta & = {PT \over PQ} \\ \sin \theta & = {PT \over 2} \\ PT & = 2\sin \theta \\ \\ \\ OR & = OS + SR \\ & = 2\sin \theta + 8\cos \theta \phantom{00} \text{ (Shown)} \end{align}

(i)

\begin{align} OR & = 2\sin \theta + 8\cos \theta \\ & = a\sin \theta + b\cos \theta \\ & = R\sin (\theta + \alpha) \\ \\ \therefore & \phantom{0} a = 2, b = 8 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2^2 + 8^2} \\ & = \sqrt{68} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {8 \over 2} \right) \\ & = 75.96^\circ \\ \\ \\ OR & = \sqrt{68} \sin (\theta + 75.96^\circ) \\ \\ \\ \text{Maximum, } OR & = \sqrt{68} (1) \\ & = \sqrt{68} \\ \\ \theta + 75.96^\circ & = 90^\circ \\ \\ \theta & = 14.04^\circ \\ & \approx 14.0^\circ \end{align}

(ii)

\begin{align} \text{When } & OR = 6, \\ 6 & = \sqrt{68} \sin (\theta + 75.96^\circ) \\ {6 \over \sqrt{68}} & = \sin (\theta + 75.96^\circ) \\ \\ \sin (\theta + 75.96^\circ) & = 0.72761 \phantom{000000} [\text{1st & 2nd quadrants since } \sin (\theta + 75.96^\circ) > 0] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.72761) \\ & = 46.69^\circ \end{align}

\begin{align} 0^\circ < \phantom{.} & \theta < 90^\circ \\ 75.96^\circ < \theta & + 75.96^\circ < 165.96^\circ \\ \\ \theta + 75.96^\circ & = 46.69^\circ \text{ (N.A.)}, 180^\circ - 46.69^\circ \\ & = 133.31^\circ \\ \\ \theta & = 57.35^\circ \\ & \approx 57.4^\circ \end{align}

(i)

\begin{align} 3\sin \theta + 4\cos (60^\circ - \theta) & = 2 \\ [\text{Addition formula: } \cos (A - B)] \phantom{000000} 3\sin \theta + 4(\cos 60^\circ \cos \theta + \sin 60^\circ \sin \theta) & = 2 \\ [\text{Special values: } \cos 60^\circ, \sin 60^\circ] \phantom{00000} 3\sin \theta + 4\left[ \left({1 \over 2}\right)\cos \theta + \left({\sqrt{3} \over 2}\right)\sin \theta \right] & = 2 \\ 3\sin \theta + 2\cos \theta + 2\sqrt{3} \sin \theta & = 2 \\ 3\sin \theta + 2\sqrt{3} \sin \theta + 2\cos \theta & = 2 \\ (3 + 2\sqrt{3})\sin \theta + 2\cos \theta & = 2 \end{align}

(ii)

\begin{align} (3 + 2\sqrt{3})\sin \theta + 2\cos \theta & = 2 \\ a\sin \theta + b\cos \theta & = 2 \\ R\sin (\theta + \alpha) & = 2 \\ \\ \therefore a = 3 + 2\sqrt{3} &, b = 2 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{(3 + 2\sqrt{3})^2 + 2^2} \\ & = \sqrt{12 + 12\sqrt{3} + 9 + 4} \\ & = \sqrt{25 + 12\sqrt{3}} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {2 \over 3 + 2\sqrt{3}} \right) \\ & = 17.19^\circ \\ \\ \\ \therefore \sqrt{25 + 12\sqrt{3}}\sin (\theta + 17.19^\circ) & = 2 \\ \sin (\theta + 17.19^\circ) & = {2 \over \sqrt{25 + 12\sqrt{3}}} \\ \sin (\theta + 17.19^\circ) & = 0.29558 \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (0.29558) \\ & = 17.19^\circ \end{align}

\begin{align} \text{Since } 0^\circ \le \phantom{.} & \theta \le 360^\circ \\ 17.19^\circ \le \theta & + 17.19^\circ \le 377.19^\circ \\ \\ \theta + 17.19^\circ & = 17.19^\circ, 180^\circ - 17.19^\circ, 17.19^\circ + 360^\circ \\ & = 17.19^\circ, 162.81^\circ, 377.19^\circ \\ \\ \theta & = 0^\circ, 145.62^\circ, 360^\circ \\ & \approx 0^\circ, 146^\circ, 360^\circ \end{align}

Show questions

\begin{align} \tan \angle PQA & = {Opp \over Adj} \\ \tan \theta & = {PQ \over QA} \\ \tan \theta & = {2 \over QA} \\ QA & = {2 \over \tan \theta} \\ & = 2\left( {1 \over \tan \theta} \right) \\ & = 2\cot \theta \\ \\ \therefore OA & = OQ + QA \\ & = 5 + 2\cot \theta \end{align}

\begin{align} \sin \angle OAM & = {Opp \over Hyp} \\ \sin \theta & = {OM \over OA} \\ \sin \theta & = {OM \over 5 + 2\cot \theta} \\ \\ OM & = \sin \theta (5 + 2\cot \theta) \\ & = \sin \theta \left[5 + 2\left({\cos \theta \over \sin \theta}\right) \right] \\ & = 5\sin \theta + 2 \cos \theta \end{align}

(i)

\begin{align} OM & = 2 \cos \theta + 5\sin \theta \\ & = a \cos \theta + b\sin \theta \\ & = R \cos (\theta - \alpha) \\ \\ \therefore & \phantom{0} a = 2, b = 5 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{2^2 + 5^2} \\ & = \sqrt{29} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {5 \over 2} \right) \\ & = 68.2^\circ \\ \\ \\ OM & = \sqrt{29} \cos (\theta - 68.2^\circ) \\ \\ \text{Maximum value of } OM & = \sqrt{29} (1) \\ & = \sqrt{29} \\ \\ \theta - 68.2^\circ & = 0^\circ, 360^\circ \\ \\ \theta & = 68.2^\circ, 428.2^\circ \text{ (Reject)} \end{align}

(ii)

\begin{align} \text{When } & OM = 4, \\ 4 & = \sqrt{29} \cos (\theta - 68.2^\circ) \\ \\ \cos (\theta - 68.2^\circ) & = {4 \over \sqrt{29}} \phantom{00000000} [ \text{1st & 4th quadrants since } \cos (\theta - 68.2^\circ) > 0 ] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left({4 \over \sqrt{29}}\right) \\ & = 42.03^\circ \end{align}

\begin{align} \text{Since } 0^\circ < \phantom{.} & \theta < 90^\circ \\ -68.2^\circ < \theta & - 68.2^\circ < 21.8^\circ \\ \\ \\ \theta - 68.2^\circ & = 42.03^\circ, 360^\circ - 42.03^\circ \\ & = 42.03^\circ \text{ (N.A.)}, 317.97^\circ \text{ (N.A.)}, -(360^\circ - 42.03^\circ), -42.03^\circ \\ & = -313.97^\circ \text{ (N.A.)}, -42.03^\circ \\ & = -42.03^\circ \\ \\ \theta & = -42.03^\circ + 68.2^\circ \\ & = 26.17^\circ \\ & \approx 26.2^\circ \end{align}

(i)

\begin{align} \text{Area of triangle } & = {1 \over 2}ab\sin C \\ \\ \text{Area of }\triangle AOB & = {1 \over 2}(4)(4)(\sin x) \\ & = 8\sin x \\ \\ \text{Area of }\triangle COD & = {1 \over 2}(6)(6)[\sin (90^\circ - x)] \\ & = 18\sin (90^\circ - x) & = 18\cos x \phantom{00000000} [\sin (90^\circ - A) = \cos A] \\ \\ S & = 8\sin x + 18\cos x \\ & = a\sin x + b\cos x \\ & = R\sin (x + \alpha) \\ \\ \therefore & \phantom{0} a = 8, b = 18 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{8^2 + 18^2} \\ & = \sqrt{388} \\ & = \sqrt{4} \times \sqrt{97} \\ & = 2\sqrt{97} \\ \\ \alpha & = \tan^{-1} \left( {b \over a} \right) \\ & = \tan^{-1} \left( {18 \over 8} \right) \\ & = 66.04^\circ \\ \\ \\ \therefore S & = 2\sqrt{97} \sin (x + 66.04^\circ) \end{align}

(ii)(a)

\begin{align} \text{Maximum, } S & = 2\sqrt{97} (1) \\ & = 2\sqrt{97} \\ \\ 90^\circ & = x + 66.04^\circ \\ \\ x & = 90^\circ - 66.04^\circ \\ & = 23.96^\circ \\ & \approx 24.0^\circ \end{align}

(ii)(b)

\begin{align} \text{Area of }\triangle COD & = 2 \times \text{Area of }\triangle AOB \\ 18\cos x & = 2 \times 8\sin x \\ 18\cos x & = 16\sin x \\ 18 & = 16{\sin x \over \cos x} \\ 18 & = 16\tan x \\ {18 \over 16} & = \tan x \\ \\ \tan x & = {9 \over 8} \phantom{000000} [\text{1st & 3rd quadrants since } \tan x > 0] \\ \\ \text{Basic angle, } \alpha & = \tan^{-1} \left( {9 \over 8} \right) \\ & = 48.36^\circ \end{align}

\begin{align} x & = 48.36^\circ, 48.36^\circ + 180^\circ \\ & = 48.36^\circ, 228.36^\circ \text{ (N.A.)} \\ & \approx 48.4^\circ \end{align}

(i)

\begin{align} p & = A\sin (\omega t) + B \sin (\omega t + c) \\ & = A\sin (\omega t) + B [\sin (\omega t) \cos c + \sin c \cos (\omega t)] \\ & = A\sin (\omega t) + B\cos c \sin (\omega t) + B\sin c \cos (\omega t) \\ & = (A + B\cos c)\sin (\omega t) + (B\sin c)\cos (\omega t) \\ & = (B \sin c)\cos (\omega t) + (A + B\cos c)\sin (\omega t) \end{align}

(ii)

\begin{align} p & = (B \sin c)\cos (\omega t) + (A + B\cos c)\sin (\omega t) \\ & = a \cos (\omega t) + b \sin (\omega t) \\ & = R \cos (\omega t - \alpha) \\ \\ \therefore & \phantom{0} a = B\sin c, b = A + B\cos c \end{align}
\begin{align} R & = \sqrt{a^2 + b^2} \\ R^2 & = (\sqrt{a^2 + b^2})^2 \\ & = a^2 + b^2 \\ & = (B\sin c)^2 + (A + B\cos c)^2 \\ & = B^2 \sin^2 c + A^2 + 2AB\cos c + B^2 \cos^2 c \\ & = B^2 \sin^2 c + B^2 \cos^2 c + A^2 + 2AB \cos c \\ & = B^2 (\sin^2 c + \cos^2 c) + A^2 + 2AB \cos c \\ & = B^2 (1) + A^2 + 2AB \cos c \phantom{00000000000000} [\text{Identity: } \sin^2 A + \cos^2 A = 1] \\ & = B^2 + A^2 + 2AB \cos c \text{ (Shown)} \end{align}

(i)

\begin{align} \text{Let } M \text{ denote the } & \text{point of intersection between } OB \text{ and } AM \\ \\ \angle OCM & = 180^\circ - 90^\circ - \theta \\ & = 90^\circ - \theta \\ \\ \angle ACB & = 90^\circ - \theta \\ \\ \angle CAB & = 180^\circ - 90^\circ - (90^\circ - \theta) \\ & = \theta \\ \\ \tan \angle CAB & = {CB \over BA} \\ \tan \theta & = {CB \over 2} \\ 2 \tan \theta & = CB \\ \\ \\ OC & = 6 - 2 \tan \theta \\ \\ \cos \angle COM & = {OM \over OC} \\ \cos \theta & = {OM \over 6 - 2 \tan \theta} \\ \cos \theta (6 - 2 \tan \theta) & = OM \\ 6 \cos \theta - 2 \cos \theta \tan \theta & = OM \\ 6 \cos \theta - 2 \cos \theta \left(\sin \theta \over \cos \theta \right) & = OM \\ 6 \cos \theta - 2 \sin \theta & = OM \\ \\ \therefore OM & = 6 \cos \theta - 2 \sin \theta \phantom{00} \text{ (Shown)} \end{align}

(ii)

\begin{align} OM & = 6 \cos \theta - 2 \sin \theta \\ & = R \cos (\theta + \alpha) \\ \\ a & = 6, b = 2 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{6^2 + 2^2} \\ & = \sqrt{40} \\ \\ \alpha & = \tan^{-1} \left(b \over a\right) \\ & = \tan^{-1} \left(2 \over 6\right) \\ & = 18.43^\circ \\ & \approx 18.4^\circ \end{align}

(iii)

\begin{align} OM & = 6 \cos \theta - 2 \sin \theta \\ & = R \cos (\theta + \alpha) \\ & = \sqrt{40} \cos (\theta + 18.43^\circ) \\ \\ \text{When } & OM = 5, \\ 5 & = \sqrt{40} \cos (\theta + 18.43^\circ) \\ {5 \over \sqrt{40}} & = \cos (\theta + 18.43^\circ) \phantom{000000} [\text{1st or 4th quadrant}] \\ \\ \text{Basic angle, } \alpha & = \cos^{-1} \left(5 \over \sqrt{40}\right) \\ & = 37.76^\circ \end{align}

\begin{align} \theta + 18.43^\circ & = 37.76^\circ, 360^\circ - 37.76^\circ \\ \\ \theta & = 19.33^\circ, 303.81^\circ \text{ (N.A.)} \\ & \approx 19.3^\circ \end{align}

(iv)

\begin{align} R & = \sqrt{40} = OA \\ \\ \alpha & = \angle AOB \end{align}

(v)

\begin{align} \sin \angle AOM & = {AM \over OA} \\ \sin (\theta + \alpha) & = {AM \over \sqrt{40}} \\ \\ AM & = \sqrt{40} \sin (\theta + \alpha) \end{align}

(vi)

\begin{align} \text{Area} & = {1 \over 2} \times OM \times AM \\ & = {1 \over 2} \times \sqrt{40} \cos (\theta + \alpha) \times \sqrt{40} \sin (\theta + \alpha) \\ & = 20 \sin (\theta + \alpha) \cos (\theta + \alpha) \\ & = 10 [ 2 \sin (\theta + \alpha) \cos (\theta + \alpha) ] \\ & = 10 \sin [2(\theta + \alpha)] \phantom{00} \text{(Shown)} \phantom{000000000000} [\sin 2A = 2 \sin A \cos A] \end{align}

(vii)

\begin{align} \text{Area} & = 10 \sin [2(\theta + \alpha)] \\ \\ \text{Max. area} & = 10 \text{ cm}^2 \\ \\ 2(\theta + \alpha) & = 90^\circ \\ \theta + \alpha & = 45^\circ \\ \theta & = 45^\circ - \alpha \\ & = 45^\circ - 18.43^\circ \\ & = 26.57^\circ \\ & \approx 26.6^\circ \end{align}

\begin{align} a & = k - 1 \\ b & = 2 + k \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{(k - 1)^2 + (2 + k)^2} \\ & = \sqrt{ k^2 - 2k + 1 + 4 + 4k + k^2 } \\ & = \sqrt{ 2k^2 + 2k + 5 } \\ \\ \\ 2k^2 + 2k + 5 & = 2 (k^2 + k) + 5 \\ & = 2 \left[ \left(k + {1 \over 2}\right)^2 - \left(1 \over 2\right)^2 \right] + 5 \phantom{000000} [\text{Complete the square}] \\ & = 2 \left(k + {1 \over 2}\right)^2 - {1 \over 2} + 5 \\ & = 2 \left(k + {1 \over 2}\right)^2 + {9 \over 2} \\ \\ \\ \text{Max. value of } P & = R \\ & = \sqrt{ 2k^2 + 2k + 5 } \\ & = \sqrt{ 2 \left(k + {1 \over 2}\right)^2 + {9 \over 2} } \\ \\ \text{Since } 2 \left(k + {1 \over 2}\right)^2 & \phantom{.} + {9 \over 2} \ge {9 \over 2}, \text{ max. value of } P \ge \sqrt{ 9 \over 2} \\ & \phantom{0000000000} \text{ max. value of } P \ge {3 \over \sqrt{2}} \end{align}