Sine, cosine and tangent | Year 9

01 / LEARN

The idea, step by step

Three ratios, three side pairs

Sine compares opposite with hypotenuse; cosine compares adjacent with hypotenuse; tangent compares opposite with adjacent. SOH–CAH–TOA helps remember the pairings after you have labelled the triangle.

The units cancel in each ratio. For an acute angle sine and cosine are between 0 and 1, while tangent can be greater than 1.

sin θ = \frac{o}{h}, cos θ = \frac{a}{h}, tan θ = \frac{o}{a}

Read the calculator correctly

Use degree mode when an angle carries a degree symbol. Enter the entire angle inside the chosen trigonometric function and retain the unrounded value for later steps.

Circle the two side labels involved. Opposite/hypotenuse selects sine, adjacent/hypotenuse selects cosine, and opposite/adjacent selects tangent.

A common trap

Use tangent whenever an angle is given.

Instead: Choose by the two side types available; an angle alone does not select tangent.

02 / WORKED EXAMPLES

See the thinking

Example 1

A known triangle

For a chosen acute angle, opposite = 7 cm, adjacent = 24 cm and hypotenuse = 25 cm. Find sine, cosine and tangent.

$sin θ = 7 / 25 = 0.28$ .
$cos θ = 24 / 25 = 0.96$ .
$tan θ = 7 / 24 \approx 0.2917$ .

0.28; 0.96; approximately 0.2917

Example 2

Evaluate an angle

Evaluate sin 40°, cos 40° and tan 40° to four decimal places.

Use degree mode.
$sin 40^{\circ} \approx 0.6428$ .
$cos 40^{\circ} \approx 0.7660$ ; $tan 40^{\circ} \approx 0.8391$ .

0.6428; 0.7660; 0.8391

Check your understanding

Which ratio connects adjacent and hypotenuse?

Show answer

Cosine

$cos θ = a / h$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Opposite = 6, adjacent = 8, hypotenuse = 10. Find sin θ.
2 marks
Using opposite = 6, adjacent = 8, hypotenuse = 10, find cos θ.
2 marks
Using opposite = 6, adjacent = 8, hypotenuse = 10, find tan θ.
2 marks
Evaluate tan 32.5° to four decimal places.
2 marks
Evaluate cos 60°.
2 marks
A student gives sin θ = 1.4 for an acute angle. Explain why this cannot be right.
3 marks

Read the worked solutions

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

Opposite = 6, adjacent = 8, hypotenuse = 10. Find sin θ.
2 marks
$sin θ = 6 / 10 = 3 / 5$ .
3/5 = 0.6
Using opposite = 6, adjacent = 8, hypotenuse = 10, find cos θ.
2 marks
$cos θ = 8 / 10 = 4 / 5$ .
4/5 = 0.8
Using opposite = 6, adjacent = 8, hypotenuse = 10, find tan θ.
2 marks
$tan θ = 6 / 8 = 3 / 4$ .
3/4 = 0.75
Evaluate tan 32.5° to four decimal places.
2 marks
Select degree mode.
$tan {32.5}^{\circ} \approx 0.6371$ .
0.6371
Evaluate cos 60°.
2 marks
$cos 60^{\circ} = 0.5$ .
0.5
A student gives sin θ = 1.4 for an acute angle. Explain why this cannot be right.
3 marks
The opposite/hypotenuse ratio must be less than 1.
The opposite side cannot exceed the hypotenuse.

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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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