KS3 Maths · Year 8 · Chapter 13: Right-angled triangles · Lesson 3

Using trigonometric ratios to find angles

Year 8About 75 minutesCalculator allowed45 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/ks3-year-8-using-trigonometric-ratios-to-find-angles/

Date

Do Now

Quick questions to start.

A1
Needed today
Round $52.456$ to $1$ decimal place.
[1 mark]
Answer
A2
Needed today
Two angles of a right-angled triangle are $9 0^{\circ}$ and $34. 6^{\circ}$ . Work out the third angle.
[1 mark]
Answer
A3
Last lesson
For angle $x$ , the opposite is $9$ cm and the adjacent is $12$ cm. Write $tan x$ as a fraction in its simplest form.
[1 mark]
Answer
A4
From chapter 9
Write $2.08 \times 1 0^{3}$ as an ordinary number.
[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 size of angle $x$ . Give your answer correct to $1$ decimal place.

Not drawn accurately

Hint The

7

cm side is opposite

x

and the

13

cm side is the hypotenuse: SOH. Use the

sin^{- 1}

button.

B1
Work out the size of angle $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$

Example 2

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

Not drawn accurately

Hint The

9

cm side touches

x

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

cos^{- 1}

B2
Work out the size of angle $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$

Example 3

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

Not drawn accurately

Hint No hypotenuse is given. Opposite and adjacent: TOA. Use

tan^{- 1}

B3
Work out the size of angle $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$

Example 4

The triangle is isosceles. Work out the size of angle $x$ and angle $y$ . Give your answers correct to $1$ decimal place.

Not drawn accurately

Hint The line of symmetry cuts the triangle into two right-angled triangles. Each has hypotenuse

9

cm and half the base,

6

cm.

B4
The triangle is isosceles. Work out the size of angle $x$ and angle $y$ . Give your answers correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$ $y = \dots \dots^{\circ}$

Practice

Year 8

C1
Use your calculator to find each angle $x$ , correct to $1$ decimal place.
(a) $sin x = 0.358$
(b) $cos x = 0.917$
(c) $tan x = 1.64$
[3 marks]
Answer(a) $\dots^{\circ}$ (b) $\dots^{\circ}$ (c) $\dots^{\circ}$
C2
Use your calculator to find each angle $x$ , correct to $1$ decimal place.
$(a) sin x = \frac{3}{8} (c) cos x = \frac{7}{12} (b) tan x = \frac{11}{4}$
[3 marks]
Answer(a) $\dots^{\circ}$ (b) $\dots^{\circ}$ (c) $\dots^{\circ}$

C3
Work out the size of angle $y$ . Give your answer correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $y = \dots \dots^{\circ}$
C5
Leo works out $sin x = \frac{13}{9}$ for an angle in a right-angled triangle. His calculator shows an error when he presses $sin^{- 1}$ . Explain why his value of $sin x$ cannot be right.
[2 marks]

Exam-style questions

GCSE Foundation

D1
Amir works out the size of angle $x$ . He writes " $tan x = \frac{7}{12}$ , so $x = 30. 3^{\circ}$ ".
Amir is wrong. Explain his mistake and work out the correct size of angle $x$ , correct to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$

D2
A ladder $6.5$ m long leans against a vertical wall. The foot of the ladder is $1.8$ m from the wall. A ladder is safe when the angle it makes with the ground is between $7 0^{\circ}$ and $8 0^{\circ}$ .
Not drawn accurately
(a)
Work out the size of angle $x$ . Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(b)
Work out how far up the wall the ladder reaches. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(c)
The foot of the ladder is moved $0.6$ m closer to the wall. Is the ladder still safe? You must show your working.
[2 marks]
Answernew angle $= \dots \dots^{\circ}$ safe: yes / no
Total for D2: 6 marks
D3
The front of a tent is an isosceles triangle. Its two sloping sides are $2.8$ m long and it is $3$ m wide at the ground.
Not drawn accurately
(a)
Work out the angle $x$ between a sloping side and the ground. Give your answer correct to $1$ decimal place.
[2 marks]
Answer
(b)
Work out the angle $y$ at the top of the tent.
[2 marks]
Answer
Total for D3: 4 marks
D4
$A$ is the point $(1, 1)$ and $B$ is the point $(9, 4)$ . Work out the angle that the line $A B$ makes with the $x$ -axis. Give your answer correct to $1$ decimal place.
[3 marks]
Answer $\dots \dots^{\circ}$

Extension

Stretch

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

E1
$A B C D$ is a rhombus. Its diagonals are $A C = 16$ cm and $B D = 9$ cm. Work out the size of each angle of the rhombus. Give your answers correct to $1$ decimal place.
[5 marks]
Not drawn accurately
Hint 1 · What to try
The diagonals of a rhombus cut each other in half at right angles. This makes four right-angled triangles.
Hint 2 · The first line
Each right-angled triangle has shorter sides $8$ cm and $4.5$ cm. Call the angle at $A$ in one of them $a$ : $tan a = \frac{4.5}{8}$ .
Hint 3 · The full method
$a = 29.3 6^{\circ}$ , so that angle of the rhombus is $2 a = 58. 7^{\circ}$ . The other angle is $180 - 58.7 = 121. 3^{\circ}$ . Two angles are $58. 7^{\circ}$ and two are $121. 3^{\circ}$ .
Answer $\dots \dots^{\circ}$ and $\dots \dots^{\circ}$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can choose sine, cosine or tangent from the two sides I know.
I can use the inverse buttons on my calculator to find an angle.
I can split an isosceles triangle into two right-angled triangles to find its angles.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers