Trigonometry

Express $6\sin\theta - 4\cos\theta$ in the form $R\sin(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.

Hence solve the equation $6\sin\theta - 4\cos\theta + 5 = 0$ for $0^\circ < \theta < 360^\circ$.

As the value of $\beta$ varies, find the greatest possible value of

and determine the smallest positive value of $\beta$, in degrees, for which this greatest value occurs.

Express $4\cos\theta\sin(\theta + 30°)$ in the form $R\cos(2\theta - \alpha) + k$, where $R > 0$, $0° < \alpha < 90°$ and $k$ is a constant.

Hence solve the equation

for $0° < \phi < 90°$.

Express $4\cos\theta\sin(\theta + 30°)$ in the form $R\cos(2\theta - \alpha) + k$, where $R > 0$, $0° < \alpha < 90°$ and $k$ is a constant.

Hence solve the equation

for $0° < \phi < 90°$.

Express $4\cos\theta\sin(\theta + 30°)$ in the form $R\cos(2\theta - \alpha) + k$, where $R > 0$, $0° < \alpha < 90°$ and $k$ is a constant.

Hence solve the equation

for $0° < \phi < 90°$.

Show that

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

Prove that $\cos(\theta + 30^\circ) \cos(\theta + 60^\circ) \equiv \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta$.

Solve the equation $5 \cos(2\alpha + 30^\circ) \cos(2\alpha + 60^\circ) = 1$ for $0^\circ < \alpha < 90^\circ$.

Show that the exact value of $\cos 20^\circ \cos 50^\circ + \cos 40^\circ \cos 70^\circ$ is $\frac{1}{2}\sqrt{3}$.