Review of radians and unit-circle values | Year 11

01 / LEARN

Understand the idea.

Radians describe arc length

A radian measures angle by arc length divided by radius. One full turn is $2 π$ radians, equivalent to $360^{\circ}$ . Convert a degree angle by multiplying by $\frac{π}{180}$ , keeping $π$ for an exact value.

The formulas $s = r θ$ and $A = \frac{1}{2} r^{2} θ$ require $θ$ in radians. Arc length has the same length units as the radius; sector area has squared units. Convert the angle before substituting the radius.

s = r θ, A = \frac{1}{2} r^{2} θ

Coordinates explain signs

The point at angle $θ$ on the unit circle is $(cos θ, sin θ)$ . Cosine is horizontal position and sine is vertical position, so the quadrant determines their signs. Tangent is their ratio wherever cosine is nonzero.

Use a first-quadrant reference angle to find exact magnitudes, then apply the actual quadrant signs. Adding $2 π$ preserves sine and cosine; adding $π$ preserves tangent. Tangent is undefined where the unit-circle horizontal coordinate is zero.

tan θ = \frac{sin θ}{cos θ}

A radius-one circle with equal coordinate scales. A first-quadrant point P has coordinates cosine theta and sine theta. The radius, angle and horizontal and vertical components are labelled.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Example 1

Find $sin (\frac{5 π}{6})$ and $cos (\frac{5 π}{6})$ exactly.

Worked solution

1
Locate the angle and reference angle.
$\frac{5 π}{6} = π - \frac{π}{6}$
The angle lies in quadrant II and has reference angle $\frac{π}{6}$ .
2
Use the positive vertical coordinate.
$sin (\frac{5 π}{6}) = sin (\frac{π}{6}) = \frac{1}{2}$
Sine is positive in quadrant II.
3
Use the negative horizontal coordinate.
$cos (\frac{5 π}{6}) = - cos (\frac{π}{6}) = - \frac{\sqrt{3}}{2}$
Cosine is negative in quadrant II.

Answer

$sin (\frac{5 π}{6}) = 1 / 2$ and $cos (\frac{5 π}{6}) = - \sqrt{3} / 2$ .

EXAMPLE 02

Example 2

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

Worked solution

1
Convert the sector angle to radians.
$θ = 135 \times \frac{π}{180} = \frac{3 π}{4}$
The arc and sector formulas use radian measure.
2
Calculate the arc length.
$s = r θ = 6 \times \frac{3 π}{4} = \frac{9 π}{2} cm$
Arc length equals radius multiplied by radian angle.
3
Calculate the sector area.
$A = \frac{1}{2} r^{2} θ = \frac{1}{2} \times 36 \times \frac{3 π}{4} = \frac{27 π}{2} {cm}^{2}$
The sector occupies the fraction $θ / (2 π)$ of the full circle area.

Answer

Arc $\frac{9 π}{2}$ cm; sector area $\frac{27 π}{2}$ cm $2$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Find $tan (\frac{7 π}{4})$ exactly.

Show answer & reasoning

$- 1$ .

$\frac{7 π}{4}$ is in quadrant IV, with sine $- \sqrt{2} / 2$ and cosine $\sqrt{2} / 2$ . Their ratio is $- 1$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

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

Read the worked solutions

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

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

In this lesson

Understand the idea.

Radians describe arc length

Coordinates explain signs

See the method in action.

Example 1

Locate the angle and reference angle.

Use the positive vertical coordinate.

Use the negative horizontal coordinate.

Example 2

Convert the sector angle to radians.

Calculate the arc length.

Calculate the sector area.

Try it yourself.

Check your understanding

Keep practising