Trigonometry

Express $5\sin(x + \frac{1}{6}\pi) - 4\cos x$ in the form $R\sin(x - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the exact value of $R$ and give the value of $\alpha$ correct to 3 decimal places.

Hence solve the equation $5\sin(2\theta + \frac{1}{6}\pi) - 4\cos 2\theta = \sqrt{7}$ for $0 \leq \theta \leq \pi$. Give your answers correct to 2 decimal places.

Express $7\sin\theta + 24\cos\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. Give the value of $\alpha$ correct to 4 decimal places.

Hence solve the equation $7\sin\frac{1}{3}x + 24\cos\frac{1}{3}x = 24.5$ for $0 < x < \pi$.

Find the exact $x$-coordinate of $M$.

By using the substitution $u = \cos x$, find the area of the region bounded by the curve, the $x$-axis between $x = 0$ and $x = \frac{1}{4}\pi$, and the line $x = \frac{1}{4}\pi$.

Prove the identity $\cot^2 \theta - \tan^2 \theta \equiv 4\cot 2\theta \csc 2\theta$.

Hence solve the equation $\cot^2 x - \tan^2 x = 5\sec 2x$ for $0^\circ < x < 90^\circ$.

Solve the differential equation, obtaining a relation between $x$ and $y$.

Given that $0 < y < \frac{1}{2}\pi$, find the values of $y$ when $x = 0$.

Express $3\sqrt{2}\sin(x + 45^\circ) + \cos x$ in the form $R\cos(x - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$.

Hence solve the equation $3\sqrt{2}\sin(3\theta + 45^\circ) + \cos 3\theta = -4$ for $0^\circ < \theta < 180^\circ$.