CAIE A-Level Mathematics 3.3 Trigonometry Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise selecting compound-, double-angle, reciprocal and R-form identities to prove results, simplify expressions and solve every valid angle in a stated interval.

Exam points

select compound- or double-angle identities that reduce the expression to one useful form
write a sin x + b cos x as R sin(x ± α) or R cos(x ± α), finding exact R and α
solve the transformed equation and generate all interval solutions without introducing extra roots

Covered syllabus points

3.3.1—Reciprocal trig functions 3.3.2—Trig identities

[Maximum number: 3]

The polynomials f(x) and g(x) are defined by

\mathrm{f}(x)=4 x^{3}+a x^{2}+8 x+15 \quad \text { and } \quad \mathrm{g}(x)=x^{2}+b x+18

where a and b are constants.

Hence solve the equation $\mathrm{f}(\operatorname{cosec} \theta)-\mathrm{g}(\operatorname{cosec} \theta)=0$ for $0<\theta<2 \pi$ .

(a)

Prove that $\cos \left(\theta+30^{\circ}\right) \cos \left(\theta+60^{\circ}\right) \equiv \frac{1}{4} \sqrt{3}-\frac{1}{2} \sin 2 \theta$ .

[ 4 ]

(b)

Solve the equation $5 \cos \left(2 \alpha+30^{\circ}\right) \cos \left(2 \alpha+60^{\circ}\right)=1$ for $0^{\circ}<\alpha<90^{\circ}$ .

[ 4 ]

(c)

Show that the exact value of $\cos 20^{\circ} \cos 50^{\circ}+\cos 40^{\circ} \cos 70^{\circ}$ is $\frac{1}{2} \sqrt{3}$ .

[ 3 ]

Exam points

Covered syllabus points

Question 6(d)

Question 7

Question 7(a)

Question 7(b)

Question 7(c)