Full Coverage: Trigonometry of Right-Angled Triangles

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"Full Coverage": Trigonometry of Right-Angled Triangles

This worksheet is designed to cover one question of each type seen in past papers, for each

GCSE Higher Tier topic. This worksheet was automatically generated by the DrFrostMaths

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Question 1 Categorisation: Determine one of the shorter sides of a right-angled triangle.

[Edexcel IGCSE June2010-3H Q7]

Work out the value of 𝑥 .

Give your answer correct to 3 significant figures.

.......................... cm

Question 2 Categorisation: As above (but using a different trigonometric ratio).


Calculate the length of LM. Give your answer correct to 3 significant figures.

.......................... cm

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Question 3 Categorisation: As above (again using a different trigonometric ratio, and with the

triangle in a different orientation)

[Edexcel IGCSE May2012-4H Q6]

Calculate the length of MN.


.......................... cm

Question 4 Categorisation: Determine the hypotenuse of a right-angled triangle.

[AQA IGCSE FM June2012-P2 Q10]

An aircraft flies 𝑦 kilometres in a straight line at an angle of elevation of 28°

The gain in height is 7 kilometres.

Work out the value of 𝑦 .

𝑦 = .......................... km


Question 5 Categorisation: Use trigonometry twice, where one triangle is embedded in another.


Triangle ABC is right-angled at B. Angle BAC = 32° AC = 47 m.

D is the point on AB such that angle BDC = 51°

Calculate the length of BD. Give your answer correct to 3 significant figures.

.......................... m

Question 6 Categorisation: As above, but for non-embedded triangles.

[Edexcel IGCSE Jan2016-4H Q19]

𝑃𝑆𝑅 is a straight line.

Angle 𝑃𝑆𝑄 = 90° 𝑃𝑆 = 8.4 cm Angle 𝑄𝑃𝑆 = 38° Angle 𝑆𝑄𝑅 = 44°

Work out the length of 𝑄𝑅. Give your answer correct to 3 significant figures.

𝑄𝑅 = .......................... cm


Question 7 Categorisation: Use a combination of Pythagoras’ theorem and trigonometry.

[Edexcel GCSE June2012-2H Q18]

The diagram shows a quadrilateral ABCD.

AB = 16 cm. AD = 12 cm. Angle BCD = 40°

Angle ADB = angle CBD = 90°

Calculate the length of CD. Give your answer correct to 3 significant figures.

.......................... cm

Question 8 Categorisation: Determine a distance in a 3D diagram using trigonometry, and

under that ‘angle of elevation’ is the angle upwards from the horizontal.


A, B and C are points on horizontal

ground. C is due West of B. A is

due South of B and AB = 40 m.

There is a vertical flagpole at B.

From A, the angle of elevation of

the top of the flagpole is 13°

From C, the angle of elevation of

the top of the flagpole is 19°

Calculate the distance AC. Give your answer correct to 3 significant figures.

.......................... m


Question 9 Categorisation: Determine an angle in a 3D diagram.

[Edexcel GCSE(9-1) Mock Set 1 Autumn 2016 3H Q19]

The diagram shows a pyramid with base 𝐴𝐵𝐶 .

𝐶𝐷 is perpendicular to both 𝐶𝐴 and 𝐶𝐵 .

Angle 𝐶𝐵𝐷 = 34° Angle 𝐴𝐷𝐵 = 45° Angle 𝐷𝐵𝐴 = 60° 𝐵𝐶 = 20 cm.

Calculate the size of the angle between the line 𝐴𝐷 and the plane 𝐴𝐵𝐶 .

Give your answer correct to 1 decimal place.

.......................... °


Question 10 Categorisation: Using trigonometry twice, both to find a side and an angle.

[Edexcel GCSE Nov2016-2H Q15]

AB = 15 m BC = 24 m Angle BAD = 62°

Work out the size of angle BCD. Give your answer correct to 1 decimal place.

.......................... °

Question 11 Categorisation: Use trigonometry in the context of circle theorems.

[Edexcel GCSE Nov2015-2H Q15b]

𝐵 , 𝐶 and 𝐷 are points on the circumference of a circle, centre 𝑂. 𝐵𝑂𝐷 is a diameter of the

circle.

Calculate the length of 𝐵𝐶 .


.......................... cm


Question 12 Categorisation: Know exact values for 𝒔𝒊𝒏, 𝒄𝒐𝒔, 𝒕𝒂𝒏 of 𝟑𝟎°, 𝟒𝟓° and 𝟔𝟎°.

[Edexcel Specimen Papers Set 1, Paper 1F Q27a, Paper 1H Q7a]

Write down the exact value of 𝑐𝑜𝑠 30°

..........................

Question 13 Categorisation: Use exact trigonometry values to find an unknown side.

Work out the length of 𝐴𝐶 .

Do not use a calculator for this question.

𝐴𝐶 = .......................... 𝑐𝑚


Question 14 Categorisation: Use exact trig values in harder multi-step problems.

[Edexcel GCSE(9-1) Mock Set 3 Autumn 2017 1H Q20 Edited]

The diagram shows three right-angled triangles.

Given that 𝑦 = 𝑘𝑛 , find the value of 𝑘 .

..........................


Question 15 Categorisation: Manipulate surds in the context of trigonometry.

[AQA IGCSE FM Jan2013-P1 Q15 Edited]

Find the value of 𝑥 .

𝑥 = .......................... °

Question 16 Categorisation: Determine an unknown angle using trigonometry.

[Edexcel GCSE Nov-2006-4I Q21, Nov2006-6H Q10]

Work out the value of 𝑥 .


.......................... °


Question 17 Categorisation: As above, but where the angle is described rather than necessarily

indicated on the diagram, e.g. “angle 𝑨𝑩𝑪”.

[Edexcel IGCSE Jan2012-3H Q9a]

Triangle PQR has a right angle at Q. PQ = 3.4 cm and PR = 5.8 cm.

Work out the size of angle QRP. Give your answer correct to 1 decimal place.

.......................... °

Question 18 Categorisation: Use trigonometry in the context of bounds.

[Edexcel IGCSE June2010-3H Q14c]

Triangle ABC is right-angled at B. AB = 20 cm, correct to 1 significant figure.

BC = 8.3 cm, correct to 2 significant figures.

Calculate the lower bound for the value of 𝑡𝑎𝑛 𝑥° .

..........................


Question 19 Categorisation: Use a combination of Pythagoras theorem and trigonometry to find

an unknown angle.

[Edexcel IGCSE Jan2014(R)-3H Q17]

The diagram shows triangle ABC.

D is the point on AB, such that CD is perpendicular to AB.

AC = 8.3 cm. AD = 4.7 cm. BD = 7.5 cm.

Calculate the size of angle ABC. Give your answer correct to 1 decimal place.

.......................... °


Question 20 Categorisation: Determine an angle of elevation.

[Edexcel IGCSE May2015(R)-3H Q23]

A, B and C are points on

horizontal ground. B is

due North of A and AB

is 14 m.

C is due East of A and

AC is 25 m. A vertical

flagpole, TX, has its

base at the point X on

BC such that the angle

AXC is a right angle.

The height of the flagpole, TX, is 10 m.

Calculate the size of the angle of elevation of T from A.


.......................... °

Question 21 Categorisation: Determine an angle in a 3D diagram.

[Edexcel IGCSE Jan2017-4H Q21b]

The diagram shows a cuboid ABCDEFGH.


AB = 21 cm and CH = 9 cm.

K is the point on EH such that angle AKB = 68° and BK = 16.5 cm.

Calculate the size of the angle between the line BK and the plane ABCD.


.......................... °

Question 22 Categorisation: As above.

[Edexcel IGCSE Jan2013-3H Q21]

The diagram shows a pyramid with a horizontal rectangular base PQRS.

PQ = 16 cm. QR = 10 cm. M is the midpoint of the line PR.

The vertex, T, is vertically above M. MT = 15 cm.

Calculate the size of the angle between TP and the base PQRS.


.......................... °


Question 23 Categorisation: Use trigonometry in the context of volumes of solids.

[Edexcel GCSE(9-1) Mock Set 3 Autumn 2017 3H Q12]

The diagram shows a metal rod, AB, resting inside a cylindrical tin.

The tin is on a horizontal table.

AC is a diameter of the base of the tin.

B is on the top edge of the tin.

BC is vertical.

The radius of the base of the tin is 5 cm.

The volume of the tin is 1178 cm3.

Find the angle between the rod and the base of the tin.

Give your answer correct to the nearest degree.

.......................... °


Question 24 Categorisation: As above.

[Edexcel IGCSE Jan2014(R)-4H Q21]

The diagram shows a solid cone.

The base of the cone is a horizontal

circle, centre O, with radius 4.5 cm.

AB is a diameter of the base and OV is

the vertical height of the cone.

The curved surface area of the cone is

130 cm2. Calculate the size of the angle

AVB. Give your answer correct to 1

decimal place.

.......................... °

Question 25 Categorisation: Use trigonometry in the context of bearings.

[Edexcel IGCSE Jan2015(R)-3H

Q11]

Town B is 35 km east and 80 km

north of town A.

Work out the bearing of town A

from town B.

Give your answer correct to the

nearest degree.

.......................... °


Answers

Question 1

5.13 cm

Question 2

6.22 cm

Question 3

105 cm

Question 4

𝑦 = 14.9 km

Question 5

20.2 m

Question 6

𝑄𝑅 = 9.12 cm

Question 7

any value in the range

16.4 cm to 16.5 cm

Question 8

48.2 m

Question 9


38.5 ° to 38.6 °

Question 10

28.9 °

Question 11


2.74 cm to 2.75 cm

Question 12

√3

2

Question 13

𝐴𝐶 = 3 𝑐𝑚

Question 14

𝑘 =3

4

Question 15

𝑥 = 60 °

Question 16

33.9 °

Question 17

35.9 °

Question 18

0.33

Question 19

42.4 °

Question 20

39.3 °

Question 21

33.1 °

Question 22

57.8 °

Question 23

56 °

Question 24

58.6 °

Question 25

204 °

Full Coverage: Trigonometry of Right-Angled Triangles

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