Exact values from special triangles | Year 10

01 / LEARN

The idea, step by step

Build the exact ratios from triangles

A 45°–45°–90° triangle has side ratio 1 : 1 : √2. A 30°–60°–90° triangle has side ratio 1 : √3 : 2, with 1 opposite 30°. Divide the appropriate sides to obtain sine, cosine and tangent.

These are exact relationships. A decimal approximation is useful for checking, but does not replace an exact answer when one is requested.

Choose the ratio before substituting

For sin 60°, use opposite over hypotenuse: √3/2. For tan 60°, use opposite over adjacent: √3. The same angle does not give the same value for every trigonometric function.

Scale a special triangle by a common factor to find lengths. Simplify surds and rationalise a denominator when that makes the exact form clearer.

A common trap

Giving tan 60° as √3/2.

Instead: That is sin 60°. Tangent uses adjacent in the denominator rather than hypotenuse.

02 / WORKED EXAMPLES

See the thinking

Example 1

Exact ratios

Find cos 30° and then simplify tan 60° × sin 30°.

In the special triangle, $cos 30^{\circ} = \sqrt{3} / 2$ .
$tan 60^{\circ} = \sqrt{3}$ and $sin 30^{\circ} = 1 / 2$ .
The product is $\sqrt{3} / 2$ .

cos 30^{\circ} = \sqrt{3} / 2

; product

\sqrt{3} / 2

Example 2

Scale a special triangle

A right triangle has hypotenuse 12 cm and an angle of 60°. Find both legs exactly.

Adjacent leg $= 12 cos 60^{\circ} = 6$ cm.
Opposite leg $= 12 sin 60^{\circ} = 6 \sqrt{3}$ cm.

6

cm and

6 \sqrt{3}

Check your understanding

Find sin 45° exactly.

Show answer

$\sqrt{2} / 2$

The 45° triangle has opposite 1 and hypotenuse √2, so $1 / \sqrt{2} = \sqrt{2} / 2$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find sin 30° exactly.
2 marks
Find cos 45° exactly.
2 marks
Find tan 30° exactly.
2 marks
A right triangle has hypotenuse 10 cm and an angle of 45°. Find the opposite leg exactly.
3 marks
Evaluate 2 cos 60° + sin 30° exactly.
2 marks
Are sin 60° and tan 60° equal? Give exact values to justify.
2 marks

Read the worked solutions

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

Find sin 30° exactly.
2 marks
The shorter leg is half the hypotenuse.
$1 / 2$
Find cos 45° exactly.
2 marks
$cos 45^{\circ} = 1 / \sqrt{2} = \sqrt{2} / 2$ .
$\sqrt{2} / 2$
Find tan 30° exactly.
2 marks
$tan 30^{\circ} = 1 / \sqrt{3} = \sqrt{3} / 3$ .
$\sqrt{3} / 3$
A right triangle has hypotenuse 10 cm and an angle of 45°. Find the opposite leg exactly.
3 marks
$x = 10 sin 45^{\circ} = 10 (\sqrt{2} / 2) = 5 \sqrt{2}$ cm.
$5 \sqrt{2}$ cm
Evaluate 2 cos 60° + sin 30° exactly.
2 marks
$2 (1 / 2) + 1 / 2 = 3 / 2$ .
$3 / 2$
Are sin 60° and tan 60° equal? Give exact values to justify.
2 marks
$sin 60^{\circ} = \sqrt{3} / 2$ whereas $tan 60^{\circ} = \sqrt{3}$ .
The ratios have different denominators.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

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