KS3 Maths · Year 8 · Chapter 13: Right-angled triangles · Lesson 1

Introduction to trigonometric ratios

Year 8About 65 minutesCalculator allowed45 marks

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Quick questions to start.

A1
Needed today
Write $\frac{5}{7}$ as a decimal correct to $3$ decimal places.
[1 mark]
Answer
A2
Needed today
Triangle $Q$ is an enlargement of triangle $P$ with scale factor $3$ . A side of $P$ is $4.5$ cm. How long is the matching side of $Q$ ?
[1 mark]
Answer
A3
Last lesson
A $400$ g jar of honey costs £ $2.60$ . Work out the cost of $100$ g.
[1 mark]
Answer
A4
From chapter 6
A right-angled triangle has shorter sides $9$ cm and $12$ cm. Work out the length of the hypotenuse.
[1 mark]
Answer

Year 8

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

Example 1

Triangle $A B C$ has a right angle at $B$ . Write down the opposite side and the adjacent side (a) for angle $A$ , (b) for angle $C$ .

Hint The hypotenuse

A C

is the same for both angles. The other two sides swap over.

B2
Triangle $P QR$ has a right angle at $Q$ . Write down the opposite side, then the adjacent side, (a) for angle $P$ , (b) for angle $R$ .
[2 marks]
Answer(a) $\dots$ , $\dots$ (b) $\dots$ , $\dots$

Example 2

For the angle marked $x$ , work out $\frac{opposite}{hypotenuse}$ , $\frac{adjacent}{hypotenuse}$ and $\frac{opposite}{adjacent}$ as decimals.

Hint Name the sides for angle

x

first: the

4.2

cm side does not touch

x

, so it is the opposite.

B3
For the angle marked $x$ , work out $\frac{opposite}{hypotenuse}$ , $\frac{adjacent}{hypotenuse}$ and $\frac{opposite}{adjacent}$ . Give each answer correct to $3$ decimal places.
[3 marks]
Answer $\dots\dots$ $\dots\dots$ $\dots\dots$

Example 3

Show that $\frac{opposite}{adjacent}$ is the same for the angle marked $x$ in both triangles. What does this tell you about the two triangles?

Not drawn accurately

Hint Work out the ratio for each triangle and compare.

B4
Work out $\frac{opposite}{hypotenuse}$ for angle $x$ and for angle $y$ . Are angles $x$ and $y$ equal? Give a reason for your answer.
[2 marks]
Not drawn accurately
Answer $\dots\dots$ $\dots\dots$ Equal? $\dots\dots$

Year 8

C1
Use your calculator to find $sin 3 8^{\circ}$ , $cos 3 8^{\circ}$ and $tan 3 8^{\circ}$ . Give each answer correct to $3$ decimal places.
[3 marks]
Answer $sin 3 8^{\circ} = \dots$ $cos 3 8^{\circ} = \dots$ $tan 3 8^{\circ} = \dots$
C2
Work out $\frac{opposite}{hypotenuse}$ for the $5 0^{\circ}$ angle, correct to $3$ decimal places. Which calculator key, $sin$ , $cos$ or $tan$ , gives the same value for $5 0^{\circ}$ ?
[2 marks]
Not drawn accurately
Answer $\dots\dots$ key $\dots\dots$

C3
Work out the length of the hypotenuse. Then work out $\frac{adjacent}{hypotenuse}$ for angle $x$ , correct to $3$ decimal places.
[3 marks]
Not drawn accurately
Answerhyp $= \dots\dots$ cm ratio $= \dots\dots$
C4
The two sides marked are used in one ratio for angle $x$ . Is it $sin x$ , $cos x$ or $tan x$ ? Give a reason.
[2 marks]
Not drawn accurately
Answer $\dots\dots$
C5
Explain why $\frac{opposite}{hypotenuse}$ is always less than $1$ for any angle in a right-angled triangle, but $\frac{opposite}{adjacent}$ can be more than $1$ .
[2 marks]
C6
For an angle in a right-angled triangle, $\frac{opposite}{adjacent} = 1$ . What can you say about the two shorter sides of the triangle? Work out the size of the angle.
[2 marks]
Answerangle $= \dots \dots^{\circ}$

GCSE Foundation

D2
Ravi says, "For angle $x$ , the $9$ cm side is the adjacent side, because it is next to the right angle."
Is Ravi correct? Explain your answer.
[2 marks]
Not drawn accurately
D3
A ladder $5.2$ m long leans against a vertical wall. The foot of the ladder is on level ground, $2$ m from the wall. The ladder makes angle $x$ with the ground.
Not drawn accurately
(a)
Work out how far up the wall the ladder reaches.
[2 marks]
Answer
(b)
Work out $\frac{opposite}{adjacent}$ for angle $x$ .
[1 mark]
Answer
(c)
A second ladder makes the same angle $x$ with the ground. Its foot is $1.5$ m from the wall. Work out how far up the wall it reaches.
[2 marks]
Answer
Total for D3: 5 marks
D4
A builder's rule says a ramp is too steep if $\frac{height it rises}{horizontal distance}$ is more than $\frac{1}{12}$ . Ramp A rises $35$ cm over a horizontal distance of $4.5$ m. Ramp B rises $0.48$ m over a horizontal distance of $5.6$ m. Show that only one of the ramps is too steep, and say which one.
[4 marks]
Not drawn accurately
AnswerToo steep: ramp $\dots\dots$

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can name the hypotenuse, opposite and adjacent sides for an angle.
I can work out a ratio of two sides as a decimal.
I can use the same ratio in similar right-angled triangles to find a side.

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