Browse Trigonometry

288 questions at your level

DifficultyAllLevel 1Level 2Level 3Level 4Level 5

The sine and cosine rules

29 questions

LessonNot started
In triangle ABCABC, angle A=40∘A = 40^{\circ}, angle B=75∘B = 75^{\circ} and a=8a = 8 cm.

Find bb, giving your answer to 33 significant figures. (3 marks)
●●●●●Level 23 marksStart

Exact values of trigonometric ratios

24 questions

LessonNot started
Write down the exact value of tan⁡45∘\tan 45^\circ.
●●●●●Level 21 markStart
Write down the exact values of sin⁡30∘\sin 30^\circ and cos⁡60∘\cos 60^\circ.
●●●●●Level 21 markStart
Solve the equation

tan⁡2x=3\tan^2 x = 3

for 0°<x<360°0° < x < 360°

(3 marks)
●●●●●Level 23 marksStart

Graphs of sine, cosine and tangent

31 questions

LessonNot started
State the amplitude and the period of the curve y=3sin⁡2x∘y = 3\sin 2x^{\circ}. (2 marks)
●●●●●Level 22 marksStart

Solving trigonometric equations

57 questions

LessonNot started
Solve the equation

tan⁡2x=3\tan^2 x = 3

for 0°<x<360°0° < x < 360°

(3 marks)
●●●●●Level 23 marksStart

Radian measure

31 questions

LessonNot started
A sector OABOAB of a circle with centre OO has radius rr cm and angle θ\theta radians. The arc ABAB has length 1212 cm and the sector has area 3636 cm2^2.
(a) Find rr and θ\theta. (4 marks)
(b) Find the area of the segment bounded by the chord ABAB and the arc ABAB, giving your answer to 3 significant figures. (2 marks)
●●●●●Level 26 marksStart
Solve 2cos⁡x=32\cos x = \sqrt{3} for 0≤x≤2π0 \le x \le 2\pi, giving your answers as exact multiples of π\pi.
●●●●●Level 23 marksStart
(a) Express 150∘150^\circ and 225∘225^\circ in radians, giving each answer as an exact multiple of π\pi.
(b) Express 7π12\frac{7\pi}{12} radians in degrees.
●●●●●Level 23 marksStart
Without using a calculator, find the exact value of (a) cos⁡5π6\cos\frac{5\pi}{6},
(b) sin⁡3π4\sin\frac{3\pi}{4},
(c) tan⁡7π6\tan\frac{7\pi}{6}.
●●●●●Level 23 marksStart

Arc length and sector area

33 questions

LessonNot started
OAB12 cm75°The diagram shows a sector OABOAB of a circle of radius 12 cm12\text{ cm} with angle AOB=75∘AOB = 75^\circ. Find the exact length of the arc ABAB.
●●●●●Level 22 marksStart
A sector OPQOPQ of a circle has centre OO and radius rr cm, where angle POQ=0.9POQ = 0.9 radians.

The area of the sector OPQOPQ is 115.2115.2 cm2^2.
(a) Show that r=16r = 16.

(2 marks)
(b) Find the perimeter of the sector OPQOPQ.

(2 marks)
●●●●●Level 24 marksStart
A sector OABOAB of a circle with centre OO has radius rr cm and angle θ\theta radians. The arc ABAB has length 1212 cm and the sector has area 3636 cm2^2.
(a) Find rr and θ\theta. (4 marks)
(b) Find the area of the segment bounded by the chord ABAB and the arc ABAB, giving your answer to 3 significant figures. (2 marks)
●●●●●Level 26 marksStart
OAB8 cm0.8 radThe diagram shows a sector OABOAB of a circle of radius 8 cm8\text{ cm}, where the angle AOB=0.8AOB = 0.8 radians. Find
(a) the length of the arc ABAB,
(b) the area of the sector.
●●●●●Level 24 marksStart
OAB6 cmθThe diagram shows a sector OABOAB of a circle of radius 6 cm6\text{ cm}, in which the arc ABAB has length 15 cm15\text{ cm}. Find
(a) the angle AOBAOB in radians,
(b) the area of the sector.
●●●●●Level 24 marksStart

Small angle approximations

23 questions

LessonNot started
Given that θ\theta is measured in radians, the value of

lim⁡θ→01−cos⁡2θθsin⁡θ\displaystyle\lim_{\theta \to 0} \dfrac{1 - \cos 2\theta}{\theta\sin\theta}

is

Circle your answer.
  • A 12\dfrac{1}{2}
  • B 11
  • C 22
  • D 44
●●●●●Level 21 markStart
θ\theta is small and measured in radians.

Using the small angle approximations for cos⁡2θ\cos 2\theta and tan⁡5θ\tan 5\theta, find an approximate value for

cos⁡2θ−1θtan⁡5θ\frac{\cos 2\theta - 1}{\theta\tan 5\theta}

(3 marks)
●●●●●Level 23 marksStart
Given that θ\theta is small and measured in radians, write down approximations for
(a) sin⁡4θ\sin 4\theta,
(b) cos⁡3θ\cos 3\theta,
(c) tan⁡2θ\tan 2\theta.
●●●●●Level 23 marksStart

Reciprocal trigonometric functions

12 questions

LessonNot started
Find the exact values of (a) sec⁡π\sec\pi,
(b) cosec⁡π2\operatorname{cosec}\frac{\pi}{2},
(c) cot⁡π3\cot\frac{\pi}{3}.
●●●●●Level 23 marksStart
Find the exact values of (a) sec⁡π4\sec\frac{\pi}{4},
(b) cosec⁡2π3\operatorname{cosec}\frac{2\pi}{3},
(c) cot⁡5π6\cot\frac{5\pi}{6}.
●●●●●Level 23 marksStart

The sec/cosec/cot identities

26 questions

LessonNot started
Write sec⁡2θ\sec^2\theta in terms of tan⁡2θ\tan^2\theta, and cosec⁡2θ\operatorname{cosec}^2\theta in terms of cot⁡2θ\cot^2\theta.
●●●●●Level 22 marksStart

Inverse trigonometric functions

12 questions

LessonNot started
State the domain and range of (a) arcsin⁡x\arcsin x,
(b) arccos⁡x\arccos x,
(c) arctan⁡x\arctan x.
●●●●●Level 23 marksStart
(a) State the range of arctan⁡x\arctan x.
(b) Explain why arctan⁡x\arctan x is defined for every real number xx, whereas arcsin⁡x\arcsin x is not.
●●●●●Level 22 marksStart
Find, in radians, the exact values of (a) arcsin⁡12\arcsin\frac{1}{2},
(b) arccos⁡0\arccos 0,
(c) arctan⁡3\arctan\sqrt{3}.
●●●●●Level 23 marksStart

Double-angle formulae

42 questions

LessonNot started
Given that θ\theta is measured in radians, the value of

lim⁡θ→01−cos⁡2θθsin⁡θ\displaystyle\lim_{\theta \to 0} \dfrac{1 - \cos 2\theta}{\theta\sin\theta}

is

Circle your answer.
  • A 12\dfrac{1}{2}
  • B 11
  • C 22
  • D 44
●●●●●Level 21 markStart

The R-form (a sinθ + b cosθ)

28 questions

LessonNot started
Given that

5sin⁡θ+3cos⁡θ≡Rsin⁡(θ+α)5\sin\theta + 3\cos\theta \equiv R\sin(\theta + \alpha)

where R>0R > 0 and 0<α<π20 < \alpha < \dfrac{\pi}{2}, find RR exactly and find α\alpha, in radians, correct to 3 decimal places.

(3 marks)
●●●●●Level 23 marksStart
Given that

4sin⁡θ−7cos⁡θ≡Rsin⁡(θ−α)4\sin\theta - 7\cos\theta \equiv R\sin(\theta - \alpha)

where R>0R > 0 and 0<α<90∘0 < \alpha < 90^\circ, find the value of RR and the value of α\alpha, giving α\alpha to 1 decimal place.

(3 marks)
●●●●●Level 23 marksStart

Proving trigonometric identities

26 questions

LessonNot started
Prove that

tan⁡θ+cot⁡θcosec⁡θ≡sec⁡θθ≠nπ2, n∈Z\frac{\tan\theta + \cot\theta}{\operatorname{cosec}\theta} \equiv \sec\theta \qquad \theta \neq \frac{n\pi}{2},\ n \in \mathbb{Z}

(3 marks)
●●●●●Level 23 marksStart