CAIE A-Level Mathematics AS 2.3 Trigonometry Questions

Practise reciprocal trigonometric functions, identities and R-formula transformations to solve equations and retain every valid angle inside specified degree or radian intervals.

Exam points

replace sec, cosec and cot with reciprocal or quotient forms and apply their identities
express a sin x + b cos x as R sin(x ± α) or R cos(x ± α) with correct quadrant
solve the reduced equation and keep all solutions lying in the specified interval

Covered syllabus points

2.3.1—Reciprocal trig functions 2.3.2—Trig identities

[Maximum number: 6]

It is given that $3 \sin 2 \theta=\cos \theta$ where $\theta$ is an angle such that $0^{\circ}<\theta<90^{\circ}$ .

(a)

Find the exact value of $\sin \theta$ .

[ 2 ]

Use $\sin 2 \theta=2 \sin \theta \cos \theta$
Obtain $\sin \theta=\frac{1}{6}$

(b)

Find the exact value of $\sec \theta$ .

[ 2 ]

Use correct identity or identities to find value of $\sec \theta$
Obtain $\frac{6}{\sqrt{35}}$ or exact equivalent

(c)

Find the exact value of $\cos 2 \theta$ .

[ 2 ]

Use correct identity or identities to find value of $\cos 2 \theta$
Obtain $\frac{17}{18}$ or exact equivalent

[Maximum number: 3]

It is given that $\theta$ is an acute angle in degrees such that $\sin \theta=\frac{2}{3}$ .
Find the exact value of $\sin \left(\theta+60^{\circ}\right)$ .

State or imply that $\cos \theta=\frac{1}{3} \sqrt{5}$
or exact equivalent.

Substitute appropriate values into $\sin \theta \cos 60^{\circ}+\cos \theta \sin 60^{\circ}$
Obtain $\frac{1}{3}+\frac{1}{6} \sqrt{15}$
or exact equivalent.

[Maximum number: 3]

The polynomial p(x) is defined by

\mathrm{p}(x)=6 x^{3}+a x^{2}-4 x-3,

where a is a constant. It is given that ( x+3 ) is a factor of p(x).

Hence solve the equation $\mathrm{p}(\operatorname{cosec} \theta)=0$ for $0^{\circ}<\theta<360^{\circ}$ .

Use the factorisation from part (b) with $x=\cosec\theta$ :
$(\cosec\theta+3)(3\cosec\theta+1)(2\cosec\theta-1)=0.$

Attempt a solution of
$\sin\theta=k,\qquad -1\le k\le1.$
The valid equation is
$\cosec\theta=-3 \quad\Rightarrow\quad \sin\theta=-\frac13.$ $\theta=199.5^\circ.$ $\theta=340.5^\circ.$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question 3