(a)

Simplify $\sin 2 \alpha \sec \alpha$ .

[ 2 ]

(b)

Given that $3 \cos 2 \beta+7 \cos \beta=0$ , find the exact value of $\cos \beta$ .

[ 3 ]

[Maximum number: 4]

Solve the equation $2 \sin \left(\theta+30^{\circ}\right)+5 \cos \theta=2 \sin \theta$ for $0^{\circ}<\theta<90^{\circ}$ .

[Maximum number: 4]

Solve the equation $\sec ^{2} \theta+\tan ^{2} \theta=5 \tan \theta+4$ for $0^{\circ}<\theta<180^{\circ}$ . Show all necessary working.

[Maximum number: 6]

By first expanding $\sin \left(\theta+30^{\circ}\right)$ , solve the equation $\sin \left(\theta+30^{\circ}\right) \operatorname{cosec} \theta=2$ for $0^{\circ}<\theta<360^{\circ}$ .

[Maximum number: 6]

Express the equation $\tan \left(\theta+45^{\circ}\right)-2 \tan \left(\theta-45^{\circ}\right)=4$ as a quadratic equation in $\tan \theta$ . Hence solve this equation for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$ .

[Maximum number: 5]

Solve the equation $5 \cos \theta(1+\cos 2 \theta)=4$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$ .

(a)

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

[ 3 ]

(b)

Hence state the greatest and least possible values of $(5 \sin x-3 \cos x)^{2}$ .

[ 2 ]

[Maximum number: 4]

Showing all necessary working, solve the equation $\sin \left(\theta-30^{\circ}\right)+\cos \theta=2 \sin \theta$ , for $0^{\circ}<\theta<180^{\circ}$ .

[Maximum number: 4]

It is given that $\sec \theta=\sqrt{17}$ where $0<\theta<\frac{1}{2} \pi$ .
Find the exact value of $\tan \left(\theta+\frac{1}{4} \pi\right)$ .

(a)

Show that the equation $\tan \left(x+45^{\circ}\right)=6 \tan x$ can be written in the form

6 \tan ^{2} x-5 \tan x+1=0

(b)

Hence solve the equation $\tan \left(x+45^{\circ}\right)=6 \tan x$ , for $0^{\circ}<x<180^{\circ}$ .

[ 3 ]