GCSE Maths · Higher · Chapter 22: Triangles

Trigonometry in 2D and 3D, trigonometric graphs, and the sine and cosine rules

Grades 4–9About 90 minutesCalculator allowed90 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
A right-angled triangle has hypotenuse 10 cm. One of its angles is $3 5^{\circ}$ . Find the length of the side opposite the $3 5^{\circ}$ angle. Give your answer to 3 significant figures.
[2 marks]
Answer

A2
The two shorter sides of a right-angled triangle are 7 cm and 24 cm. Find the length of the hypotenuse.
[1 mark]
Answer
A3
In a right-angled triangle, the side opposite angle $θ$ is 5 cm and the side next to it is 8 cm. Find $θ$ to 1 decimal place.
[2 marks]
Answer
A4
Write down the exact value of $sin 3 0^{\circ}$ .
[1 mark]
Answer
A5
A triangle has base 9 cm and perpendicular height 6 cm. Find its area.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

In triangle $A B C$ , angle $A = 4 8^{\circ}$ , angle $B = 6 7^{\circ}$ and $B C = 12$ cm. Find the length of $A C$ . Give your answer to 3 significant figures.

Hint Each side is opposite an angle. Use

\frac{A C}{s i n B} = \frac{B C}{s i n A}

B1
Find the length of $P R$ . Give your answer to 3 significant figures.
[3 marks]
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Answer

Example 2

In triangle $A B C$ , $A B = 11$ cm, $A C = 8$ cm and angle $B A C = 4 0^{\circ}$ . Find the length of $B C$ . Give your answer to 3 significant figures.

Hint You know two sides and the angle between them. Use

a^{2} = b^{2} + c^{2} - 2 b c cos A

B2
Find the length of $L N$ . Give your answer to 3 significant figures.
[3 marks]
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Answer

Example 3

A triangle has sides 7 cm, 9 cm and 12 cm. Find the size of its largest angle. Give your answer to 1 decimal place.

Hint The largest angle is opposite the longest side. Rearrange the cosine rule:

cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}

B3
A triangle has sides 6 cm, 10 cm and 13 cm. Find the size of its largest angle. Give your answer to 1 decimal place.
[3 marks]
Answer

Example 4

Find the area of a triangle with sides 14 cm and 9 cm and an angle of $3 8^{\circ}$ between them. Give your answer to 3 significant figures.

Hint Area

= \frac{1}{2} ab sin C

, where

C

is the angle between sides

a

and

b

B4
Find the area of the triangle. Give your answer to 3 significant figures.
[2 marks]
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Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Answer these without drawing a graph. Use the symmetry of the sine and cosine graphs.
(a)
$sin 3 4^{\circ} = 0.559$ to 3 decimal places. Write down the other angle between $0^{\circ}$ and $18 0^{\circ}$ whose sine is $0.559$ .
[1 mark]
Answer
(b)
Write down the angle $x$ between $18 0^{\circ}$ and $36 0^{\circ}$ for which $cos x = cos 7 0^{\circ}$ .
[1 mark]
Answer
(c)
Write down the coordinates of the lowest point on the graph of $y = cos x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
Total for C1: 3 marks

C2
The graph of $y = sin x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ is drawn with the line $y = - 0.4$ .
Solve $sin x = - 0.4$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give your answers to 1 decimal place.
[3 marks]
Answer
C3
In triangle $A B C$ , $B C = 9$ cm, $A C = 7$ cm and angle $B A C = 5 0^{\circ}$ . Find the size of angle $A B C$ . Give your answer to 1 decimal place.
[3 marks]
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Answer
C4
The diagram shows a cuboid $A B C D E F G H$ . $A B = 8$ cm, $B C = 6$ cm and $A E = 5$ cm.
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(a)
Find the length of the diagonal $A G$ . Give your answer to 3 significant figures.
[2 marks]
Answer
(b)
Find the angle between $A G$ and the base $A B C D$ . Give your answer to 1 decimal place.
[2 marks]
Answer
Total for C4: 4 marks
C5
The area of the triangle is 30 cm². Find the value of $x$ . Give your answer to 3 significant figures.
[3 marks]
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Answer
C6
Triangle $P QR$ is isosceles with $P Q = P R = 9$ cm and angle $QP R = 10 4^{\circ}$ . Find the length of $QR$ . Give your answer to 3 significant figures.
[3 marks]
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Answer
C7
$V A B C D$ is a pyramid with a square base of side 6 cm. The vertex $V$ is directly above $M$ , the centre of the base, and $V M = 8$ cm.
Find the angle between the edge $V A$ and the base. Give your answer to 1 decimal place.
[3 marks]
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Answer
C8
Two sides of a triangle are 7 cm and 10 cm. The angle between them is $12 0^{\circ}$ . Find the perimeter of the triangle. Give your answer to 3 significant figures.
[3 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
A boat sails 12 km from $A$ to $B$ on a bearing of $05 0^{\circ}$ . It then sails 9 km from $B$ to $C$ on a bearing of $16 0^{\circ}$ .
Work out the distance $A C$ . Give your answer to 3 significant figures.
[4 marks]
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Answer

D2
Ravi says, "The equation $sin x = 0.5$ has only one solution between $0^{\circ}$ and $36 0^{\circ}$ . It is $x = 3 0^{\circ}$ ."
Ravi is wrong. Explain why, and give all the solutions between $0^{\circ}$ and $36 0^{\circ}$ .
[2 marks]
Answer
D3
In triangle $A B C$ , $A B = 11$ cm, $B C = 8$ cm and angle $B A C = 4 2^{\circ}$ .
Two different triangles fit this information. Find the two possible sizes of angle $A C B$ . Give your answers to 1 decimal place.
[3 marks]
Answer
D4
The diagram shows a quadrilateral $A B C D$ . $A B = 8$ cm, $B C = 11$ cm, angle $A B C = 7 5^{\circ}$ , angle $C A D = 4 2^{\circ}$ and angle $A D C = 6 3^{\circ}$ .
Not drawn accurately
(a)
Work out the length of $A C$ . Give your answer to 3 significant figures.
[2 marks]
Answer
(b)
Work out the length of $C D$ . Give your answer to 3 significant figures.
[2 marks]
Answer
(c)
Work out the area of $A B C D$ . Give your answer to 3 significant figures.
[2 marks]
Answer
Total for D4: 6 marks
D5
$P R S T$ is a horizontal rectangular field with $P R = 20$ m and $P T = 15$ m. A vertical flagpole $P Q$ of height 9 m stands at the corner $P$ .
Find the angle of elevation of $Q$ from the opposite corner $S$ . Give your answer to 1 decimal place.
[3 marks]
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Answer
D6
A triangle has sides 8 cm and 9 cm. Its area is 25 cm².
Find the two possible sizes of the angle between the 8 cm and 9 cm sides. Give your answers to 1 decimal place.
[3 marks]
Answer
D7
A triangle has sides 7 cm, 8 cm and 13 cm.
Show that its largest angle is $12 0^{\circ}$ .
[3 marks]
D8
Solve $3 cos x + 1 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give your answers to 1 decimal place.
[3 marks]
Answer
D9
In triangle $A B C$ , angle $A = 3 5^{\circ}$ , angle $B = 8 5^{\circ}$ and $A B = 9$ cm.
Work out the area of the triangle. Give your answer to 3 significant figures.
[4 marks]
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Answer
D10
$A$ and $B$ are points on level ground in a straight line with the foot $F$ of a vertical tower $F T$ . $A B = 15$ m. The angle of elevation of $T$ from $A$ is $3 2^{\circ}$ and from $B$ is $4 8^{\circ}$ .
Work out the height of the tower. Give your answer to 3 significant figures.
[4 marks]
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Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
In triangle $A B C$ , $A B = 23$ cm, $A C = 5$ cm and angle $B A C = 3 0^{\circ}$ .
Find the exact length of $B C$ .
[3 marks]
Hint 1 · What to try
You know two sides and the angle between them, and $cos 3 0^{\circ}$ has an exact value.
Hint 2 · The first line
$B C^{2} = (23)^{2} + 5^{2} - 2 \times 23 \times 5 \times \frac{3}{2}$
Hint 3 · The full method
$B C^{2} = 12 + 25 - 30 = 7$ , so $B C = 7$ cm.
Answer

E2
Find every value of $x$ between $0^{\circ}$ and $36 0^{\circ}$ for which $sin x = cos x$ .
[3 marks]
Hint 1 · What to try
Sketch both graphs on the same axes and look for where they cross.
Hint 2 · The first line
They cross where $tan x = 1$ , which is at $4 5^{\circ}$ .
Hint 3 · The full method
The graphs repeat the crossing $18 0^{\circ}$ later, where both are negative. So $x = 4 5^{\circ}$ or $x = 22 5^{\circ}$ .
Answer
E3
In triangle $A B C$ , $A B = 12$ cm, angle $B A C = 3 0^{\circ}$ and $B C = k$ cm.
Find the range of values of $k$ for which two different triangles can be drawn.
[3 marks]
Hint 1 · What to try
Draw $A B$ and the line from $A$ at $3 0^{\circ}$ . Then swing an arc of radius $k$ from $B$ and see how many times it meets the line.
Hint 2 · The first line
The shortest distance from $B$ to the line is $12 sin 3 0^{\circ} = 6$ .
Hint 3 · The full method
The arc meets the line twice on the correct side of $A$ when it is longer than 6 but shorter than $A B$ . So $6 < k < 12$ .
Answer
E4
Find the angle between a diagonal of a cube and one of its faces. Give your answer to 1 decimal place.
[3 marks]
Hint 1 · What to try
Call the side of the cube 1. Find the length of the diagonal of a face.
Hint 2 · The first line
The face diagonal is $2$ , and the cube's diagonal rises 1 above it.
Hint 3 · The full method
$tan θ = \frac{1}{2}$ , so $θ = 35. 3^{\circ}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use trigonometry and Pythagoras in several steps, in 2D and 3D, and find the angle between a line and a plane.
I can use the sine and cosine graphs to find the second solution a calculator does not give.
I can use the sine rule and the cosine rule to find a side or an angle, and choose between them.
I can find the area of any triangle with $\frac{1}{2} ab sin C$ .

Answers and mark scheme

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