CAIE A-Level Mathematics 1.5 Trigonometry Question Bank

CAIE A-Level Mathematics 1.5 Trigonometry Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise sketching trigonometric graphs, proving identities and solving equations in stated intervals using exact values, inverse functions, periodicity and all valid branches.

Exam points

  • sketch sine, cosine or tangent transformations with the correct period, range and key points
  • prove identities by transforming one side with reciprocal, quotient and Pythagorean identities
  • solve the reduced trigonometric equation and generate every solution in the stated interval

Covered syllabus points

1.5.1—Trig graphs1.5.2—Exact trig values1.5.3—Inverse trig notation1.5.4—Trig identities1.5.5—Trig equations

Question 1

[Maximum number: 4]

Solve the equation 6 sin(theta)=1+2/sin(theta) for -180 degrees < theta < 180 degrees.

Question 2

Question 2(a)

(a)

Solve the equation tan⁡−1(5x−3)=−14π\tan ^{-1}(5 x-3)=-\frac{1}{4} \pi.

[ 2 ]

Question 2(b)

(b)

Solve the equation 5cos⁡2θ=4sin⁡θ+45 \cos ^{2} \theta=4 \sin \theta+4 for 12π⩽θ⩽2π\frac{1}{2} \pi \leqslant \theta \leqslant 2 \pi.

[ 4 ]

Question 5

Question 5(a)

(a)

Show that cos⁡4x+2sin⁡2x−1≡8sin⁡4x−6sin⁡2x\cos 4 x+2 \sin ^{2} x-1 \equiv 8 \sin ^{4} x-6 \sin ^{2} x.

[ 4 ]

Question 5(b)

(b)

Hence solve the equation cos⁡4x+2sin⁡2x−1=0\cos 4 x+2 \sin ^{2} x-1=0 for −180∘⩽x⩽180∘-180^{\circ} \leqslant x \leqslant 180^{\circ}.

[ 4 ]

Question 3(a)

[Maximum number: 3]

The function f is defined by f(x)=tan⁡2(12x)\mathrm{f}(x)=\tan ^{2}\left(\frac{1}{2} x\right) for 0⩽x<π0 \leqslant x<\pi.

Find the exact value of f′(23π)\mathrm{f}^{\prime}\left(\frac{2}{3} \pi\right).

Question 5

[Maximum number: 4]

The equation of a curve is y = 4cos(2x) + 3 for 0 <= x <= 2π.

Question 5(a)

(a)

State the greatest and least possible values of y.

[ 2 ]

Question 5(b)

(b)

Sketch the curve.

[ 2 ]