Express $24 \sin \theta-7 \cos \theta$ in the form $R \sin (\theta-\alpha)$ , where R>0 and $0^{\circ}<\alpha<90^{\circ}$ . Give the value of $\alpha$ correct to 2 decimal places.

[ 3 ]

(b)

Hence find the smallest positive value of $\theta$ satisfying the equation

24 \sin \theta-7 \cos \theta=17

[Maximum number: 5]

Solve the equation $\cos \left(\theta-60^{\circ}\right)=3 \sin \theta$ , for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$ .

[Maximum number: 7]

It is given that $\cos a=\frac{3}{5}$ , where $0^{\circ}<a<90^{\circ}$ . Showing your working and without using a calculator to evaluate a,

(a)

find the exact value of $\sin \left(a-30^{\circ}\right)$ ,

[ 3 ]

(b)

find the exact value of $\tan 2 a$ , and hence find the exact value of $\tan 3 a$ .

[ 4 ]

(a)

Show that $(\sec x+\cos x)^{2}$ can be expressed as $\sec ^{2} x+a+b \cos 2 x$ , where a and b are constants to be determined.

[ 2 ]

(a)

Express $8 \cos \theta+15 \sin \theta$ in the form $R \cos (\theta-\alpha)$ , where R>0 and $0^{\circ}<\alpha<90^{\circ}$ . Give the value of $\alpha$ correct to 2 decimal places.

(b)

Hence solve the equation $8 \cos \theta+15 \sin \theta=12$ , giving all solutions in the interval $0^{\circ}<\theta<360^{\circ}$ .

(a)

Using the expansions of $\cos (3 x+x)$ and $\cos (3 x-x)$ , show that

\frac{1}{2}(\cos 4 x+\cos 2 x) \equiv \cos 3 x \cos x

[ 3 ]

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