Right-triangle revision | Year 11

01 / LEARN

The idea, step by step

Label before calculating

The hypotenuse is opposite the right angle and is the longest side. Opposite and adjacent depend on the acute angle you choose. Here adjacent means the adjacent leg, not the hypotenuse. Label the angle first, then choose the ratio that contains the given and required sides.

sin θ = o / h, cos θ = a / h, tan θ = o / a

Solve and check

Rearrange before rounding. For an unknown angle, use the inverse function in the correct calculator mode. Keep extra digits until the final answer. An acute angle must be between 0° and 90°, and a hypotenuse must exceed either leg. Use these facts to detect an incorrect ratio or mode.

A common trap

Using the hypotenuse in tangent.

Instead: Tangent compares the two legs only; identify the hypotenuse separately.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

A 5 m ladder makes an angle of $60^{\circ}$ with level ground. Find its vertical height.

Height is opposite; the ladder is the hypotenuse.
$sin 60^{\circ} = h / 5$ , so $h = 5 sin 60^{\circ}$ .
Use the exact ratio before rounding.

5 \sqrt{3} / 2 m \approx 4.33 m

Example 2

Worked example 2

The legs opposite and adjacent to theta are 6 cm and 8 cm. Find theta to one decimal place.

$tan θ = 6 / 8 = 0.75$ .
$θ = {tan}^{- 1} (0.75)$ in degree mode.
The result is acute, as required.

{36.9}^{\circ}

Check your understanding

Opposite is 5 and hypotenuse is 13. Find $sin θ$ .

Show answer

$5 / 13$

Sine is opposite divided by hypotenuse.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A right triangle has legs 9 cm and 12 cm. For the angle opposite 9 cm, find all three ratios.
3 marks
Find the opposite side when the hypotenuse is 14 m and the angle is $30^{\circ}$ .
3 marks
The adjacent leg is 8 cm and the angle is $60^{\circ}$ . Find the hypotenuse.
3 marks
A ramp rises 1.2 m over a horizontal run of 4 m. Find its angle to one decimal place.
3 marks
A wire makes $35^{\circ}$ with horizontal ground and reaches 10 m horizontally. Find its length to two decimal places.
3 marks
Explain why a right triangle cannot have hypotenuse 7 cm and leg 9 cm.
3 marks

Read the worked solutions

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

A right triangle has legs 9 cm and 12 cm. For the angle opposite 9 cm, find all three ratios.
3 marks
$h = \sqrt{9^{2} + 12^{2}} = 15$ .
Substitute into the definitions and simplify.
$sin θ = 3 / 5, cos θ = 4 / 5, tan θ = 3 / 4$
Find the opposite side when the hypotenuse is 14 m and the angle is $30^{\circ}$ .
3 marks
$o = 14 sin 30^{\circ} = 7$ .
7 m
The adjacent leg is 8 cm and the angle is $60^{\circ}$ . Find the hypotenuse.
3 marks
$cos 60^{\circ} = 8 / h$ , so $h = 16$ .
16 cm
A ramp rises 1.2 m over a horizontal run of 4 m. Find its angle to one decimal place.
3 marks
$tan θ = 1.2 / 4 = 0.3$ .
$θ = {tan}^{- 1} (0.3) \approx {16.699}^{\circ}$ .
${16.7}^{\circ}$
A wire makes $35^{\circ}$ with horizontal ground and reaches 10 m horizontally. Find its length to two decimal places.
3 marks
$L = 10 / cos 35^{\circ} \approx 12.2077$ .
12.21 m
Explain why a right triangle cannot have hypotenuse 7 cm and leg 9 cm.
3 marks
A sine or cosine ratio of 9/7 would exceed 1, confirming inconsistent side data.
The hypotenuse must be longest.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

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