Sine, cosine and tangent | Year 9

01 / LEARN

Understand the idea.

Three ratios, three side pairs

After choosing an acute angle and labelling its sides, select a ratio: sine uses opposite over hypotenuse, cosine uses adjacent over hypotenuse, and tangent uses opposite over adjacent. SOH–CAH–TOA records these three pairings.

Both lengths in a ratio must use the same units, which then cancel. The hypotenuse is longer than either leg, so acute-angle sine and cosine lie strictly between zero and one. Tangent may be greater than one.

\begin{aligned} sin θ & = \frac{o}{h} \\ cos θ & = \frac{a}{h} \\ tan θ & = \frac{o}{a} \end{aligned}

Read the calculator correctly

Choose the function from the two side types involved, rather than from the presence of an angle alone. If opposite and adjacent are involved, use tangent; if the hypotenuse is involved, choose sine or cosine according to the other side.

For angles given in degrees, set the calculator to degree mode and enter the full angle inside the function. Retain the calculator value during further calculations, and round only the requested final result.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A known triangle

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

Worked solution

1
Use opposite over hypotenuse for sine.
$sin θ = \frac{7}{25} = 0.28$
Sine compares the side across from the angle with the longest side.
2
Use adjacent over hypotenuse for cosine.
$cos θ = \frac{24}{25} = 0.96$
Cosine uses the non-hypotenuse side touching the angle.
3
Use opposite over adjacent for tangent.
$tan θ = \frac{7}{24} \approx 0.2917$
The tangent ratio does not use the hypotenuse and is rounded here to four decimal places.

Answer

0.28; 0.96; approximately 0.2917

EXAMPLE 02

Evaluate an angle

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

Worked solution

1
Set the calculator to degree mode.
$θ = 40^{\circ}$
The degree symbol specifies the angle unit required by the question.
2
Evaluate sine and round.
$\begin{aligned} sin (40^{\circ}) & = 0.642787 \dots \\ \approx 0.6428 \end{aligned}$
The fifth decimal digit rounds the fourth decimal digit up.
3
Evaluate cosine and round.
$\begin{aligned} cos (40^{\circ}) & = 0.766044 \dots \\ \approx 0.7660 \end{aligned}$
The trailing zero records four decimal places.
4
Evaluate tangent and round.
$\begin{aligned} tan (40^{\circ}) & = 0.839099 \dots \\ \approx 0.8391 \end{aligned}$
Use the full calculator result before rounding to the requested precision.

Answer

0.6428; 0.7660; 0.8391

03 / YOUR TURN

Try it yourself.

Check your understanding

Which ratio connects adjacent and hypotenuse?

Show answer & reasoning

Cosine

Cosine connects the adjacent side and hypotenuse: $cos θ = \frac{a}{h}$ . Neither sine nor tangent uses this exact pair.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 $tan {32.5}^{\circ}$ to four decimal places.
2 marks
Evaluate $cos 60^{\circ}$ .
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
$sin θ = \frac{6}{10} = \frac{3}{5}$ .
$\frac{3}{5} = 0.6$
Using opposite $= 6$ , adjacent $= 8$ , hypotenuse $= 10$ , find $cos θ$ .
2 marks
$cos θ = \frac{8}{10} = \frac{4}{5}$ .
$\frac{4}{5} = 0.8$
Using opposite $= 6$ , adjacent $= 8$ , hypotenuse $= 10$ , find $tan θ$ .
2 marks
$tan θ = \frac{6}{8} = \frac{3}{4}$ .
$\frac{3}{4} = 0.75$
Evaluate $tan {32.5}^{\circ}$ to four decimal places.
2 marks
Select degree mode.
$tan {32.5}^{\circ} \approx 0.6371$ .
0.6371
Evaluate $cos 60^{\circ}$ .
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 $\frac{opposite}{hypotenuse}$ 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 ↗

In this lesson

Understand the idea.

Three ratios, three side pairs

Read the calculator correctly

See the method in action.

A known triangle

Use opposite over hypotenuse for sine.

Use adjacent over hypotenuse for cosine.

Use opposite over adjacent for tangent.

Evaluate an angle

Set the calculator to degree mode.

Evaluate sine and round.

Evaluate cosine and round.

Evaluate tangent and round.

Try it yourself.

Check your understanding

Keep practising