KS3 Maths · Year 8 · Chapter 13: Right-angled triangles · Lesson 4

Using trigonometric ratios to find lengths

Year 8About 75 minutesCalculator allowed47 marks

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Date

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Quick questions to start.

A1
Needed today
Solve $\frac{x}{5} = 0.84$ .
[1 mark]
Answer
A2
Needed today
Solve $\frac{12}{x} = 0.75$ .
[1 mark]
Answer
A3
Last lesson
In a right-angled triangle, the side opposite angle $x$ is $6$ cm and the hypotenuse is $12$ cm. Write down the size of angle $x$ .
[1 mark]
Answer
A4
From chapter 10
A cylinder has radius $3$ cm and height $10$ cm. Write its volume in terms of $π$ .
[1 mark]
Answer

Example, then your turn

Year 8

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

Example 1

Work out the length $x$ . Give your answer correct to $1$ decimal place.

Not drawn accurately

Hint

x

is opposite the angle and

11

cm is the hypotenuse: SOH.

sin 3 4^{\circ} = \frac{x}{11}

, so multiply both sides by

11

B1
Work out the length $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Example 2

Work out the length $x$ . Give your answer correct to $1$ decimal place.

Not drawn accurately

Hint

x

touches the angle and the right angle, so it is the adjacent. Adjacent and hypotenuse: CAH.

cos 2 3^{\circ} = \frac{x}{14}

B2
Work out the length $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Example 3

Work out the length $x$ . Give your answer correct to $1$ decimal place.

Not drawn accurately

Hint No hypotenuse is used. Opposite (

x

) and adjacent (

9

cm): TOA.

tan 4 0^{\circ} = \frac{x}{9}

B3
Work out the length $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Example 4

Work out the length $x$ . Give your answer correct to $1$ decimal place.

Not drawn accurately

Hint

sin 3 5^{\circ} = \frac{8}{x}

. This time

x

is on the bottom, so you divide:

x = \frac{8}{sin 3 5 ^{\circ}}

B4
Work out the length $x$ . Give your answer correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Practice

Year 8

C1
Work out the length of the hypotenuse $x$ . Give your answer correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm
C2
In a right-angled triangle the two acute angles are $2 0^{\circ}$ and $7 0^{\circ}$ . Explain why $sin 2 0^{\circ} = cos 7 0^{\circ}$ .
[2 marks]

C3
Work out the perimeter of this right-angled triangle. Give your answer correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $\dots\dots$ cm
C4
The triangle is isosceles. Work out the length of the base $x$ . Give your answer correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Exam-style questions

GCSE Foundation

D1
Zara works out the length $x$ . She writes " $x = 6.5 \times sin 3 6^{\circ} = 3.8$ cm".
Zara is wrong. Explain how you can tell without working anything out. Then work out the correct length of $x$ , correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

D2
A zip wire is $110$ m long. It goes down from the top of a tower in a straight line at $1 0^{\circ}$ below the horizontal.
Not drawn accurately
(a)
Work out the vertical drop of the zip wire. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(b)
Work out the horizontal distance the zip wire covers. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(c)
A second zip wire has the same vertical drop, but it goes down at $7^{\circ}$ below the horizontal. Work out its length.
[2 marks]
Answer
Total for D2: 6 marks
D3
Kai stands on level ground $25$ m from the foot of a vertical tree. The angle from the ground up to the top of the tree is $3 2^{\circ}$ .
Not drawn accurately
(a)
Work out the height of the tree. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(b)
Kai walks towards the tree until the angle up to the top of the tree is $5 0^{\circ}$ . How far from the tree is he now? Give your answer correct to $1$ decimal place.
[3 marks]
Answer
Total for D3: 5 marks
D4
The diagram shows a parallelogram with sides $11$ cm and $6$ cm. One of its angles is $6 2^{\circ}$ .
Not drawn accurately
(a)
Work out the height of the parallelogram. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(b)
Work out the area of the parallelogram. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
Total for D4: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
An equilateral triangle has a height of $9$ cm. Work out its perimeter and its area. Give your answers correct to $1$ decimal place.
[5 marks]
Hint 1 · What to try
Every angle of an equilateral triangle is $6 0^{\circ}$ . The height cuts it into two right-angled triangles.
Hint 2 · The first line
In one of them the height is opposite the $6 0^{\circ}$ angle and a side $s$ of the equilateral triangle is the hypotenuse: $sin 6 0^{\circ} = \frac{9}{s}$ .
Hint 3 · The full method
$s = 9 \div sin 6 0^{\circ} = 10.39 \dots$ cm. Perimeter $= 3 \times 10.39 \dots = 31.2$ cm. Area $= \frac{1}{2} \times 10.39 \dots \times 9 = 46.8$ cm².
Answerperimeter $= \dots\dots$ cm area $= \dots\dots$ cm²

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can choose sine, cosine or tangent from the side I know and the side I want.
I can multiply when the unknown side is on the top and divide when it is on the bottom.
I can use a length I have found to work out a perimeter or an area.

Answers and mark scheme

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