Finding an angle | Year 9 | Ascent Library

01 / LEARN

The idea, step by step

Work backwards from a ratio

Two known sides give a side ratio. Inverse sine, cosine or tangent then gives the angle with that ratio. Write the fraction first, then apply the inverse function to the entire fraction.

On a calculator sin⁻¹, cos⁻¹ and tan⁻¹ mean inverse functions here, not reciprocals such as 1/sin.

θ = {tan}^{- 1} (\frac{o}{a})

The answer must fit the triangle

Use degrees and check for an answer between 0° and 90°. A sine or cosine input outside −1 to 1 is impossible; in this acute-angle setting it lies strictly between 0 and 1.

If opposite exceeds adjacent, the angle exceeds 45°. The other acute angle equals 90° minus the angle you found.

A common trap

Use 1/tan(0.75) to invert a tangent ratio.

Instead: Use the inverse-tangent command. A reciprocal calculates a different number.

02 / WORKED EXAMPLES

See the thinking

Example 1

Opposite and adjacent

Opposite = 9 cm and adjacent = 12 cm. Find θ.

$tan θ = 9 / 12 = 0.75$ .
$θ = {tan}^{- 1} (0.75) \approx 36.8699^{\dots \circ}$ .

36.87°

Example 2

Adjacent and hypotenuse

Adjacent = 7 m and hypotenuse = 10 m. Find θ.

$cos θ = 7 / 10$ .
$θ = {cos}^{- 1} (0.7) \approx 45.5730^{\dots \circ}$ .

45.57°

Check your understanding

Opposite and adjacent are equal. Find θ.

Show answer

45°

$tan θ = 1$ , so $θ = 45^{\circ}$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find θ if o = 5 cm and h = 13 cm. Give two decimal places.
2 marks
Find θ if a = 12 cm and h = 13 cm. Give two decimal places.
2 marks
Find θ if o = 15 m and a = 8 m. Give two decimal places.
2 marks
An acute angle has sine 0.8. Find it to two decimal places.
2 marks
For hypotenuse 13 and opposite 5, a student enters sin⁻¹(13/5). Explain the error and correct it.
3 marks
One acute angle in a right triangle is 34.7°. Find the other.
2 marks

Read the worked solutions

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

Find θ if o = 5 cm and h = 13 cm. Give two decimal places.
2 marks
$θ = {sin}^{- 1} (5 / 13)$ .
22.62°
Find θ if a = 12 cm and h = 13 cm. Give two decimal places.
2 marks
$θ = {cos}^{- 1} (12 / 13)$ .
This agrees with the matching 5–12–13 triangle.
22.62°
Find θ if o = 15 m and a = 8 m. Give two decimal places.
2 marks
$θ = {tan}^{- 1} (15 / 8)$ .
Opposite exceeds adjacent, so θ exceeds 45°.
61.93°
An acute angle has sine 0.8. Find it to two decimal places.
2 marks
$θ = {sin}^{- 1} (0.8)$ .
53.13°
For hypotenuse 13 and opposite 5, a student enters sin⁻¹(13/5). Explain the error and correct it.
3 marks
The side ratio was reversed.
13/5 exceeds 1 and cannot be a sine value.
Use ${sin}^{- 1} (5 / 13)$
One acute angle in a right triangle is 34.7°. Find the other.
2 marks
$90^{\circ} - {34.7}^{\circ} = {55.3}^{\circ}$ .
55.3°

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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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