Unit-circle meaning | Year 10

01 / LEARN

The idea, step by step

Coordinates extend the triangle ratios

Start at (1,0) and turn anticlockwise through θ. On a circle of radius 1, the point has coordinates (cos θ, sin θ). In the first quadrant this agrees with the usual right triangle. The coordinate definition still works beyond 90°.

Cosine is horizontal and sine is vertical. Their signs follow the point: both positive in quadrant I; cosine negative and sine positive in II; both negative in III; cosine positive and sine negative in IV.

Use a reference angle and a quadrant

A reference angle is the acute angle to the x-axis. For 150°, it is 30°, so the coordinate magnitudes are √3/2 and 1/2. The quadrant supplies the signs.

One full turn is 360°. Adding whole turns changes the angle label but not the point. Tangent is sin θ divided by cos θ, so it is undefined wherever cos θ is zero.

A common trap

Using a positive sine value in every quadrant.

Instead: Sine is the y-coordinate. It is negative below the x-axis.

02 / WORKED EXAMPLES

See the thinking

Example 1

An obtuse angle

Find the exact unit-circle coordinates at 150°.

The reference angle is $180^{\circ} - 150^{\circ} = 30^{\circ}$ .
Quadrant II has negative x and positive y.
Therefore $(cos 150^{\circ}, sin 150^{\circ}) = (- \sqrt{3} / 2, 1 / 2)$ .

(- \sqrt{3} / 2, 1 / 2)

Example 2

Recover an angle from coordinates

A unit-circle point is (−3/5,−4/5). Find its quadrant and angle between 0° and 360° to 0.1°.

${(- 3 / 5)}^{2} + {(- 4 / 5)}^{2} = 1$ , so the point is on the unit circle.
Both coordinates are negative, so it is in quadrant III.
The reference angle is ${tan}^{- 1} (4 / 3) = 53.130^{\dots \circ}$ .
Add 180°.

Quadrant III; 233.1°

Check your understanding

Find the point at 90°.

Show answer

(0, 1)

The radius points straight up: cosine is zero and sine is one.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Give the exact unit-circle coordinates at 60°.
2 marks
Find sin 210° exactly.
2 marks
Find cos 240° exactly.
2 marks
Find tan 135° exactly.
2 marks
Do 45° and 765° identify the same unit-circle point? Explain.
2 marks
For a quadrant-II angle, cos θ = −3/5. Find sin θ and tan θ exactly.
3 marks

Read the worked solutions

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

Give the exact unit-circle coordinates at 60°.
2 marks
Coordinates are $(cos 60^{\circ}, sin 60^{\circ})$ .
$(1 / 2, \sqrt{3} / 2)$
Find sin 210° exactly.
2 marks
210° is in quadrant III with reference angle 30°.
Sine is negative, so $sin 210^{\circ} = - 1 / 2$ .
$- 1 / 2$
Find cos 240° exactly.
2 marks
The reference angle is 60° in quadrant III.
$cos 240^{\circ} = - cos 60^{\circ} = - 1 / 2$ .
$- 1 / 2$
Find tan 135° exactly.
2 marks
$tan 135^{\circ} = (\sqrt{2} / 2) / (- \sqrt{2} / 2) = - 1$ .
$- 1$
Do 45° and 765° identify the same unit-circle point? Explain.
2 marks
765° − 45° = 720°, which is two complete turns.
Yes.
For a quadrant-II angle, cos θ = −3/5. Find sin θ and tan θ exactly.
3 marks
${sin}^{2} θ = 1 - 9 / 25 = 16 / 25$ .
Sine is positive in quadrant II.
$tan θ = (4 / 5) / (- 3 / 5) = - 4 / 3$ .
$sin θ = 4 / 5$ ; $tan θ = - 4 / 3$

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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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