Unit-circle definitions and exact values | Year 11

01 / LEARN

The idea, step by step

Coordinates define the ratios

The point reached by a turn θ on a unit circle is $(cos θ, sin θ)$ . Cosine is horizontal; sine is vertical. This extends the ratios beyond acute angles. The circle equation gives ${sin}^{2} θ + {cos}^{2} θ = 1$ . Tangent is sine divided by cosine, where cosine is nonzero.

Reference angles and signs

Find the acute angle to the horizontal axis, use a 30–60–90 or 45–45–90 triangle for the magnitudes, then attach coordinate signs. Left of the origin gives negative cosine; below the horizontal axis gives negative sine. This coordinate method explains, rather than merely memorises, quadrant signs.

A common trap

Giving all three ratios the same sign in a quadrant.

Instead: Use coordinate signs first; tangent inherits the sign of sine divided by cosine.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find sine, cosine and tangent of $5 π / 6$ .

The reference angle isπ/6 in quadrant II.
Sine is positive; cosine is negative.
Divide sine by cosine for tangent.

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

Example 2

Worked example 2

Find $sin (- π / 4)$ and $cos (- π / 4)$ .

The clockwise turn reaches quadrant IV.
Both reference magnitudes are√2/2.
The vertical coordinate is negative and horizontal positive.

- \sqrt{2} / 2

and

\sqrt{2} / 2

Check your understanding

What unit-circle point corresponds to θ=π?

Show answer

$(- 1, 0)$

A half-turn reaches the leftmost point.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $sin (7 π / 6)$ exactly.
3 marks
Find $cos (3 π / 4)$ exactly.
3 marks
Find $tan (5 π / 3)$ exactly.
3 marks
Find $sin (11 π / 6)$ and $cos (11 π / 6)$ .
3 marks
If $cos θ = - 3 / 5$ in quadrant III, find sine and tangent.
3 marks
Explain why $tan (π / 2)$ is undefined.
3 marks

Read the worked solutions

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

Find $sin (7 π / 6)$ exactly.
3 marks
Reference angleπ/6; quadrant III has negative sine.
$- 1 / 2$
Find $cos (3 π / 4)$ exactly.
3 marks
Reference angleπ/4; the horizontal coordinate is negative.
$- \sqrt{2} / 2$
Find $tan (5 π / 3)$ exactly.
3 marks
Reference angleπ/3 in quadrant IV; divide negative sine by positive cosine.
$- \sqrt{3}$
Find $sin (11 π / 6)$ and $cos (11 π / 6)$ .
3 marks
Reference angleπ/6 in quadrant IV.
$- 1 / 2, \sqrt{3} / 2$
If $cos θ = - 3 / 5$ in quadrant III, find sine and tangent.
3 marks
${sin}^{2} θ = 1 - 9 / 25 = 16 / 25$ .
Select the negative sine; divide to obtain tangent.
$sin θ = - 4 / 5, tan θ = 4 / 3$
Explain why $tan (π / 2)$ is undefined.
3 marks
$cos (π / 2) = 0$ ; tangent is sine divided by cosine.
It would require division by zero.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

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