Trigonometry

Find the perimeter of the shaded region.

Find the area of the shaded region.

Show that $3\tan^2\theta + 5\sin^2\theta \equiv \frac{8\sin^2\theta - 5\sin^4\theta}{1 - \sin^2\theta}$.

Hence solve the equation $3\tan^2\theta + 5\sin^2\theta = 9$ for $0^\circ < \theta < 270^\circ$.

It is given that the area of the triangle $ABC$ is $4\text{ cm}^2$ and the area of the sector $ABC$ is $8\alpha\text{ cm}^2$.

Find the exact area of the shaded segment.

It is given instead that the length of the chord $BC$ is $\frac{1}{\sqrt{2}}r\text{ cm}$ but the area of the triangle $ABC$ is still $4\text{ cm}^2$.

Find the area of the shaded segment. Give your answer correct to 3 significant figures.

State the greatest and least possible values of $y$.

Sketch the curve.

Hence determine the number of solutions of the equation $4\cos 2x + 3 = 2x - 1$ for $0 \le x \le 2\pi$.

Prove the identity

Hence solve the equation

for $0^\circ \le \theta \le 180^\circ$.

Show that $\tan^4 \theta - 1 \equiv \frac{1 - 2\cos^2 \theta}{\cos^4 \theta}$.

Hence solve the equation $\cos^2 \theta (\tan^4 \theta - 1) = 7$ for $0^\circ < \theta < 180^\circ$.

Sketch the graph of $y = 3\sin x + 2$ for $0 \le x \le 2\pi$.

$3\sin x + 2 = x$

$3\sin x + 2 = 5 - x$

Solve the equation $3\sin x + 2 = 5\cos^2 x - 1$ for $0 \le x \le 2\pi$.