Finding a side | Year 9 | Ascent Library

01 / LEARN

The idea, step by step

Label, choose, substitute

Mark the acute angle, label opposite/adjacent/hypotenuse, and name the unknown x. Choose the ratio containing the known side and x; you do not have to calculate all three sides.

If x is the numerator, multiplication often isolates it. If x is the denominator, multiply both sides by x before dividing. Writing the equation avoids reversing the ratio.

Check the size

A calculated hypotenuse must be longer than either leg. A leg found from a hypotenuse must be shorter. A small acute angle has a relatively short opposite side.

Keep the trigonometric value unrounded until the final step. The numerical examples here use degrees and report lengths to two decimal places.

A common trap

For cos 42° = 9/x, use x = 9 cos 42°.

Instead: The unknown is in the denominator: rearrange to x = 9/cos 42°.

02 / WORKED EXAMPLES

See the thinking

Example 1

Unknown opposite side

Hypotenuse = 12 cm and θ = 35°. Find opposite.

$sin 35^{\circ} = x / 12$ .
$x = 12 sin 35^{\circ} \approx 6.8829 \dots$ .

6.88 cm

Example 2

Unknown hypotenuse

Adjacent = 9 m and θ = 42°. Find the hypotenuse.

$cos 42^{\circ} = 9 / x$ .
$x cos 42^{\circ} = 9$ , so $x = 9 / cos 42^{\circ}$ .

12.11 m

Check your understanding

Hypotenuse = 10 and θ = 30°. Find opposite.

Show answer

$o = 10 sin 30^{\circ} = 5$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find opposite if h = 14 cm and θ = 28°. Give two decimal places.
2 marks
Find adjacent if h = 18 m and θ = 57°. Give two decimal places.
2 marks
Find opposite if a = 11 cm and θ = 35°. Give two decimal places.
2 marks
Find adjacent if o = 8 m and θ = 41°. Give two decimal places.
2 marks
Find hypotenuse if a = 12 cm and θ = 27°. Give two decimal places.
2 marks
Find hypotenuse if o = 9 m and θ = 63°. Give two decimal places.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find opposite if h = 14 cm and θ = 28°. Give two decimal places.
2 marks
$o = 14 sin 28^{\circ}$ .
Evaluate in degrees and round the final length.
6.57 cm
Find adjacent if h = 18 m and θ = 57°. Give two decimal places.
2 marks
$a = 18 cos 57^{\circ}$ .
The result is shorter than 18 m.
9.80 m
Find opposite if a = 11 cm and θ = 35°. Give two decimal places.
2 marks
$o = 11 tan 35^{\circ}$ .
7.70 cm
Find adjacent if o = 8 m and θ = 41°. Give two decimal places.
2 marks
$tan 41^{\circ} = 8 / a$ .
$a = 8 / tan 41^{\circ}$ .
9.20 m
Find hypotenuse if a = 12 cm and θ = 27°. Give two decimal places.
2 marks
$cos 27^{\circ} = 12 / h$ .
$h = 12 / cos 27^{\circ}$ .
13.47 cm
Find hypotenuse if o = 9 m and θ = 63°. Give two decimal places.
2 marks
$sin 63^{\circ} = 9 / h$ .
$h = 9 / sin 63^{\circ}$ .
10.10 m

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Mathematics

Year 9 similarity, Pythagoras and right-triangle trigonometric ratios, with prerequisite revision. Non-right-triangle sine/cosine rules are not required.

Official curriculum source ↗

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