\begin{align*} a \cos x - b \sin x & = R \cos (x + \alpha) \\ 3 \cos x - 4 \sin x & = R \cos (x + \alpha) \\ \\ a = 3, & \phantom{.} 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 \\ \\ \therefore 3 \cos x - 4 \sin x & = 5 \cos (x + 53.13^\circ) \\ & \approx 5 \cos (x + 53.1^\circ) \end{align*}

\begin{align*} 3 \cos x - 4 \sin x & = R \cos (x + \alpha) \\ 3 \cos x - 4 \sin x & = R (\cos x \cos \alpha - \sin x \sin \alpha) \phantom{000000} [\text{Formula: } \cos (A + B)] \\ 3 \cos x - 4 \sin x & = R\cos x \cos \alpha - R \sin x \sin \alpha \\ 3 \cos x - 4 \sin x & = (R \cos \alpha) \cos x - (R \sin \alpha) \sin x \\ \\ 3 & = R \cos \alpha \phantom{000} \text{--- (1)} \\ \\ 4 & = R \sin \alpha \phantom{000} \text{--- (2)} \\ \\ \\ (2) & \div (1), \\ {4 \over 3} & = {R \sin \alpha \over R \cos \alpha} \\ {4 \over 3} & = {\sin \alpha \over \cos \alpha} \\ {4 \over 3} & = \tan \alpha \\ \tan^{-1} \left(4 \over 3\right) & = \alpha \\ 53.13^\circ & = \alpha \\ \\ \\ (1)^2 & + (2)^2, \\ 3^2 + 4^2 & = (R \cos \alpha)^2 + (R \sin \alpha)^2 \\ 25 & = R^2 \cos^2 \alpha + R^2 \sin^2 \alpha \\ 25 & = R^2 (\cos^2 \alpha + \sin^2 \alpha) \\ 25 & = R^2 (1) \phantom{000000} [\sin^2 A + \cos^2 A = 1] \\ 25 & = R^2 \\ \sqrt{25} & = R \\ 5 & = R \\ \\ \\ \therefore 3 \cos x - 4 \sin x & = 5 \cos (x + 53.13^\circ) \\ & \approx 5 \cos (x + 53.1^\circ) \end{align*}

\begin{align*} a \sin x - b \cos x & = R \sin (x - \alpha) \\ \sqrt{6} \sin x - \cos x & = R \sin (x - \alpha) \\ \\ a = \sqrt{6}, & \phantom{.} b = 1 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{(\sqrt{6})^2 + (1)^2} \\ & = \sqrt{7} \\ \\ \alpha & = \tan^{-1} \left(b \over a\right) \\ & = \tan^{-1} \left(1 \over \sqrt{6}\right) \\ & = 22.207^\circ \\ \\ \therefore \sqrt{6} \sin x - \cos x & = \sqrt{7} \sin (x - 22.207^\circ) \\ \\ \sqrt{6} \sin x - \cos x + 3 & \approx \sqrt{7} \sin (x - 22.2^\circ) + 3 \end{align*}

\begin{align*} \sqrt{6} \sin x - \cos x + 3 & = \underbrace{\sqrt{7} \sin (x - 22.207^\circ)}_{ -\sqrt{7} \le \sqrt{7} \sin (x - 22.207^\circ) \le \sqrt{7} } + 3 \\ \\ \text{Maximum value} & = \sqrt{7} + 3 \\ \\ \text{Let } \sqrt{7} \sin (x - 22.207^\circ) & = \sqrt{7} \\ \sin (x - 22.207^\circ) & = {\sqrt{7} \over \sqrt{7}} \\ & = 1 \phantom{000000} [\text{1st or 2nd quadrant}] \\ \\ \text{Basic angle, } \alpha & = \sin^{-1} (1) \\ & = 90^\circ \\ \\ x - 22.207^\circ & = 90^\circ, 180^\circ - 90^\circ \\ & = 90^\circ \\ \\ x & = 90^\circ + 22.207^\circ \\ & = 112.207^\circ \\ & \approx 112.2^\circ \end{align*}

\begin{align*} (\sqrt{6} \sin x - \cos x)^2 + 3 & = [ \underbrace{\sqrt{7} \sin (x - 22.207^\circ)}_{ -\sqrt{7} \le \sqrt{7} \sin (x - 22.207^\circ) \le \sqrt{7} } ]^2 + 3 \\ \\ \text{Maximum value} & = (\sqrt{7})^2 + 3 \phantom{000000} [\text{Same answer if you use } - \sqrt{7}] \\ & = 10 \\ \\ \text{Minimum value} & = (0)^2 + 3 \\ & = 3 \end{align*}

\begin{align*} (\sqrt{6} \sin x - \cos x + 3 )^2 & = [ \underbrace{\sqrt{7} \sin (x - 22.207^\circ)}_{ -\sqrt{7} \le \sqrt{7} \sin (x - 22.207^\circ) \le \sqrt{7} } + 3 ]^2 \\ \\ \text{Maximum value} & = (\sqrt{7} + 3 )^2 \\ & = (\sqrt{7})^2 + 2(\sqrt{7})(3) + (3)^2 \phantom{000000} [(a + b)^2 = a^2 + 2ab + b^2 ] \\ & = 7 + 6 \sqrt{7} + 9 \\ & = 16 + 6 \sqrt{7} \\ \\ \text{Minimum value} & = (-\sqrt{7} + 3)^2 \\ & = (3 - \sqrt{7})^2 \\ & = (3)^2 - 2(3)(\sqrt{7}) + (\sqrt{7})^2 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2 ] \\ & = 9 - 6 \sqrt{7} + 7 \\ & = 16 - 6 \sqrt{7} \end{align*}

\begin{align*} (\sqrt{6} \sin x - \cos x)^3 + 3 & = [ \underbrace{\sqrt{7} \sin (x - 22.207^\circ)}_{ -\sqrt{7} \le \sqrt{7} \sin (x - 22.207^\circ) \le \sqrt{7} } ]^3 + 3 \\ \\ \text{Maximum value} & = (\sqrt{7})^3 + 3 \\ & = 7 \sqrt{7} +3 \phantom{000000.} [ (\sqrt{7})^3 = \sqrt{7} \times \sqrt{7} \times \sqrt{7}] \\ \\ \text{Minimum value} & = (-\sqrt{7})^3 + 3 \\ & = -7 \sqrt{7} + 3 \phantom{00000} [ (- \sqrt{7})^3 = - \sqrt{7} \times - \sqrt{7} \times -\sqrt{7} ] \end{align*}

\begin{align*} a \sin \theta - b \cos \theta & = R \sin (\theta - \alpha) \\ \\ a = 0.5, b & = 0.1 \\ \\ R & = \sqrt{a^2 + b^2} \\ & = \sqrt{0.5^2 + 0.1^2} \\ & = \sqrt{0.26} \\ \\ \alpha & = \tan^{-1} \left(b \over a\right) \\ & = \tan^{-1} \left(0.1 \over 0.5\right) \\ & = 11.310^\circ \\ \\ \therefore h & = 0.5 \sin \theta - 0.1 \cos \theta \\ & = \sqrt{0.26} \sin (\theta - 11.31^\circ) \\ & \approx \sqrt{0.26} \sin (\theta - 11.3^\circ) \end{align*}