KS3 Maths · Year 8 · Chapter 13: Right-angled triangles · Lesson 2

Finding trigonometric ratios of angles

Year 8About 70 minutesCalculator allowed45 marks

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Quick questions to start.

A1
Needed today
Write $\frac{24}{40}$ in its simplest form.
[1 mark]
Answer
A2
Needed today
Write $\frac{9}{16}$ as a decimal.
[1 mark]
Answer
A3
Last lesson
For angle $x$ in a right-angled triangle, the opposite is $9$ cm and the hypotenuse is $12$ cm. Work out opposite $\div$ hypotenuse.
[1 mark]
Answer
A4
From chapter 12
A car travels $150$ km in $2.5$ hours. Work out its average speed.
[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

For the angle marked $x$ , write $sin x$ , $cos x$ and $tan x$ as fractions.

Not drawn accurately

Hint

sin = \frac{opp}{hyp}

cos = \frac{adj}{hyp}

tan = \frac{opp}{adj}

(SOH CAH TOA).

B1
For the angle marked $x$ , write $sin x$ , $cos x$ and $tan x$ as fractions.
[3 marks]
Not drawn accurately
Answer $sin x = \dots$ $cos x = \dots$ $tan x = \dots$

Example 2

Two sides are given for angle $x$ . Which ratio do they give: $sin x$ , $cos x$ or $tan x$ ? Work it out as a decimal.

Not drawn accurately

Hint The

12

cm side touches

x

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

B2
Two sides are given for angle $x$ . Which ratio do they give: $sin x$ , $cos x$ or $tan x$ ? Work it out as a decimal.
[2 marks]
Not drawn accurately
Answer $\dots\dots x = \dots\dots$

Example 3

For an angle $x$ in a right-angled triangle, $cos x = \frac{5}{13}$ . Sketch the triangle and work out $sin x$ and $tan x$ as fractions.

Not drawn accurately

Hint

cos = \frac{adj}{hyp}

, so draw the adjacent as

5

and the hypotenuse as

13

. Then use Pythagoras' theorem.

B3
For an angle $x$ in a right-angled triangle, $sin x = \frac{33}{65}$ . Sketch the triangle and work out $cos x$ and $tan x$ as fractions.
[3 marks]
Answer $cos x = \dots\dots$ $tan x = \dots\dots$

Example 4

Write down $sin x$ , $cos y$ , $tan x$ and $tan y$ . What do you notice?

Not drawn accurately

Hint The side opposite

x

is the side adjacent to

y

B4
Angles $x$ and $y$ are in the same right-angled triangle. $sin x = \frac{8}{17}$ and $tan x = \frac{8}{15}$ . Write down $cos y$ and $tan y$ .
[2 marks]
Answer $cos y = \dots\dots$ $tan y = \dots\dots$

Practice

Year 8

C1
The sides of this triangle are $p$ , $q$ and $r$ . Write $sin x$ , $cos x$ and $tan x$ using these letters.
[3 marks]
Answer $sin x = \dots$ $cos x = \dots$ $tan x = \dots$
C2
Work out the length of the hypotenuse. Then write $cos x$ as a fraction.
[3 marks]
Not drawn accurately
Answerhyp $= \dots$ cm $cos x = \dots$

C3
$sin 3 0^{\circ} = \frac{1}{2}$ . Use this to work out the side marked ?.
[2 marks]
Not drawn accurately
Answer $\dots\dots$ cm
C4
$tan 4 5^{\circ} = 1$ . Use this to work out the side marked ?. Give a reason.
[2 marks]
Not drawn accurately
Answer $\dots\dots$ cm

Exam-style questions

GCSE Foundation

D1
Mia says, "For angle $x$ , $sin x = \frac{28}{45}$ , because sine is the opposite over the adjacent."
Is Mia correct? Give the correct value of $sin x$ .
[2 marks]
Not drawn accurately
Answer $sin x = \dots\dots$

D2
The diagram shows the end of a roof. It is $8.4$ m wide and $4$ m high in the middle. The dashed line is the line of symmetry.
Not drawn accurately
(a)
Work out the length of one sloping side of the roof.
[2 marks]
Answer
(b)
Work out $sin x$ and $cos x$ . Give each answer correct to $3$ decimal places.
[2 marks]
Answer $sin x = \dots$ $cos x = \dots$
(c)
Write down $cos y$ . Give a reason for your answer.
[1 mark]
Answer $cos y = \dots$
Total for D2: 5 marks
D3
A rope $22.5$ m long goes from the top of a vertical mast to a point on level ground. The rope makes angle $x$ with the ground, and $sin x = 0.6$ .
Not drawn accurately
(a)
Work out the height of the mast.
[2 marks]
Answer
(b)
Work out how far the rope is fixed from the foot of the mast.
[2 marks]
Answer
(c)
Write $tan x$ as a fraction in its simplest form.
[1 mark]
Answer
Total for D3: 5 marks
D4
You are told that $cos 6 0^{\circ} = \frac{1}{2}$ . An equilateral triangle has sides of $10$ cm. It is cut along its line of symmetry into two right-angled triangles.
(a)
Explain why one angle of each right-angled triangle is $6 0^{\circ}$ . Then use $cos 6 0^{\circ} = \frac{1}{2}$ to show that the shortest side of each right-angled triangle is $5$ cm.
[2 marks]
(b)
Work out the height of the equilateral triangle. 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
In this right-angled triangle, $tan x = 1.875$ . The perimeter of the triangle is $60$ cm. Work out the area of the triangle.
[5 marks]
Hint 1 · What to try
Write $1.875$ as a fraction: $tan x = \frac{15}{8}$ . Call the sides $15 k$ and $8 k$ .
Hint 2 · The first line
The hypotenuse is $(15 k)^{2} + (8 k)^{2} = 17 k$ , so the perimeter is $8 k + 15 k + 17 k = 40 k = 60$ .
Hint 3 · The full method
$k = 1.5$ : the shorter sides are $12$ cm and $22.5$ cm, so the area is $\frac{1}{2} \times 12 \times 22.5 = 135$ cm².
Answer $\dots\dots$ cm²

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write sine, cosine and tangent of an angle as fractions of the sides.
I can choose which ratio two given sides make.
I can go from one ratio back to the triangle and find the other ratios.

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