Review of radians and unit-circle values | Year 11

01 / LEARN

The idea, step by step

Radians describe arc length

One radian cuts off an arc equal in length to the radius. A complete turn is $2 π$ radians, so multiply degrees by $π / 180$ to convert to radians. Retain $π$ for an exact answer rather than immediately rounding it.

For a radius $r$ and radian angle $θ$ , the arc length is $r θ$ and sector area is $r^{2} θ / 2$ . Both formulas depend on using radians. The units of the first result are length; those of the second are area.

s = r θ, A = \frac{1}{2} r^{2} θ

Coordinates explain signs

On a radius-one circle, the point at angle $θ$ is $(cos θ, sin θ)$ . The horizontal coordinate gives cosine, the vertical coordinate gives sine, and their ratio gives tangent when cosine is not zero.

Find a first-quadrant reference angle, then choose signs from the actual quadrant. Angles differing by $2 π$ share sine and cosine. Tangent repeats every $π$ and is undefined at odd multiples of $π / 2$ .

tan θ = \frac{sin θ}{cos θ}

A common trap

Substituting the degree number straight into the arc formula.

Instead: Convert the angle to radians before using $s = r θ$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Find $sin (5 π / 6)$ and $cos (5 π / 6)$ exactly.

The reference angle is $π / 6$ , and the angle is in quadrant II.
Sine is positive and cosine negative.
Use the special-angle magnitudes $1 / 2$ and $\sqrt{3} / 2$ .

sin (5 π / 6) = 1 / 2

and

cos (5 π / 6) = - \sqrt{3} / 2

Example 2

A radius-6 cm circle has a sector angle of $135^{\circ}$ . Find the arc length and sector area.

Convert $135^{\circ}$ to $3 π / 4$ radians.
The arc is $6 (3 π / 4) = 9 π / 2$ cm.
The area is $\frac{1}{2} (36) (3 π / 4) = 27 π / 2$ cm $2$ .

Arc

9 π / 2

cm; sector area

27 π / 2

2

Check your understanding

Find $tan (7 π / 4)$ exactly.

Show answer

$- 1$ .

The reference angle is $π / 4$ ; sine is negative and cosine positive.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Convert $210^{\circ}$ to radians.
2 marks
Convert $5 π / 4$ radians to degrees.
2 marks
Find $sin (4 π / 3)$ exactly.
2 marks
Find $cos (- π / 3)$ exactly.
2 marks
Find the arc length for radius 8 m and angle $3 π / 8$ .
2 marks
Explain why tangent is undefined at $π / 2$ .
2 marks

Read the worked solutions

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

Convert $210^{\circ}$ to radians.
2 marks
$210 (π / 180) = 7 π / 6$ .
$7 π / 6$ .
Convert $5 π / 4$ radians to degrees.
2 marks
$(5 π / 4) (180 / π) = 225$ .
$225^{\circ}$ .
Find $sin (4 π / 3)$ exactly.
2 marks
Quadrant III gives negative sine; the reference angle is $π / 3$ .
$- \sqrt{3} / 2$ .
Find $cos (- π / 3)$ exactly.
2 marks
Cosine is even, so this equals $cos (π / 3)$ .
$1 / 2$ .
Find the arc length for radius 8 m and angle $3 π / 8$ .
2 marks
$s = 8 (3 π / 8) = 3 π$ .
$3 π$ m.
Explain why tangent is undefined at $π / 2$ .
2 marks
$sin (π / 2) = 1$ , $cos (π / 2) = 0$ ; division by zero is undefined.
Its cosine denominator is zero.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

Official curriculum source ↗

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