[Maximum number: 3]

Solve the equation $4 \sin \theta+\tan \theta=0$ for $0^{\circ}<\theta<180^{\circ}$ .

[Maximum number: 3]

Solve the equation $8 \sin ^{2} \theta+6 \cos \theta+1=0$ for $0^{\circ}<\theta<180^{\circ}$ .

[Maximum number: 4]

Solve the equation $2 \cos \theta=7-\frac{3}{\cos \theta}$ for $-90^{\circ}<\theta<90^{\circ}$ .

[Maximum number: 4]

Given that $\theta$ is an obtuse angle measured in radians and that $\sin \theta=k$ , find, in terms of k, an expression for

(a)

$\cos \theta$ ,

[ 1 ]

(b)

$\tan \theta$ ,

[ 2 ]

(c)

$\sin (\theta+\pi)$ .

[ 1 ]

[Maximum number: 2]

The diagram shows part of the graph of $y=a+b \sin x$ . State the values of the constants a and b.

[Maximum number: 3]

Given that $\cos x=p$ , where x is an acute angle in degrees, find, in terms of p,

(a)

$\sin x$ ,

[ 1 ]

(b)

$\tan x$ ,

[ 1 ]

(c)

$\tan \left(90^{\circ}-x\right)$ .

[ 1 ]

(a)

Prove the identity $\tan ^{2} \theta-\sin ^{2} \theta \equiv \tan ^{2} \theta \sin ^{2} \theta$ .

[ 3 ]

(b)

Use this result to explain why $\tan \theta>\sin \theta$ for $0^{\circ}<\theta<90^{\circ}$ .

[ 1 ]

(a)

Show that the equation

3(2 \sin x-\cos x)=2(\sin x-3 \cos x)

can be written in the form $\tan x=-\frac{3}{4}$ .

[ 2 ]

(b)

Solve the equation $3(2 \sin x-\cos x)=2(\sin x-3 \cos x)$ , for $0^{\circ} \leqslant x \leqslant 360^{\circ}$ .

[ 2 ]

[Maximum number: 4]

The acute angle x radians is such that $\tan x=k$ , where k is a positive constant. Express, in terms of k,

(a)

$\tan (\pi-x)$ ,

[ 1 ]

(b)

$\tan \left(\frac{1}{2} \pi-x\right)$ ,

[ 1 ]

(c)

$\sin x$ .

[ 2 ]

[Maximum number: 3]

The diagram shows part of the curve with equation $y=p \sin (q \theta)+r$ , where p, q and r are constants.

(a)

State the value of p.

[ 1 ]

(b)

State the value of q.

[ 1 ]

(c)

State the value of r.

[ 1 ]