10. Radians and the Unit Circle | Mathematics Advanced

Starting point: Years 9 and 10

Circles

For a circle of radius $r$ :

$Circumference = 2 π r Area = π r^{2}$

Arcs and sectors in degrees

An arc is a fraction of the circumference. A sector is the same fraction of the area. For an angle of $θ^{\circ}$ at the centre:

$arc length = \frac{θ}{360} \times 2 π r sector area = \frac{θ}{360} \times π r^{2}$

Angles bigger than 90 degrees

On a set of axes, measure angles anticlockwise from the positive x-axis. The four quarters are called quadrants, numbered 1 to 4 anticlockwise. Angles in quadrant 2 are between $9 0^{\circ}$ and $18 0^{\circ}$ .

Angles and circles

Degree formulas

Arc: θ/360 × 2πr
Sector: θ/360 × πr²

Radians

π rad = 180°
Arc l = rθ
Sector A = ½r²θ

Unit circle

Radius 1
x = cos θ
y = sin θ

Signs

All, Sin, Tan, Cos
Related angle

Year 11Radians

One radian is the angle that makes an arc equal to the radius. A full turn is $2 π$ radians, so

$π radians = 18 0^{\circ}$

To convert degrees to radians, multiply by $\frac{π}{180}$ . To convert back, multiply by $\frac{180}{π}$ .

Degrees	$3 0^{\circ}$	$4 5^{\circ}$	$6 0^{\circ}$	$9 0^{\circ}$	$18 0^{\circ}$	$36 0^{\circ}$
Radians	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$π$	$2 π$

With $θ$ in radians, the formulas lose the 360:

$l = r θ A = \frac{1}{2} r^{2} θ$

Ruby's slice: $θ = \frac{π}{4}$ , so $l = 20 \times \frac{π}{4} = 5 π$ . Same answer, less work.

Segment area. The region between a chord and its arc is a sector minus a triangle:

$A = \frac{1}{2} r^{2} (θ - sin θ)$

Year 11The unit circle

The unit circle has centre $(0, 0)$ and radius 1. Draw an angle $θ$ from the positive x-axis. The point where it meets the circle is

$P = (cos θ, sin θ)$

This defines sine and cosine for every angle, and $tan θ = \frac{sin θ}{cos θ}$ .

The point on the circle has coordinates (cos θ, sin θ). The letters show which ratios are positive in each quadrant.

ASTC ("All Stations To Central") tells you which ratios are positive:

Quadrant 1: All positive.
Quadrant 2: only Sin positive.
Quadrant 3: only Tan positive.
Quadrant 4: only Cos positive.

Finding any exact value. Find the related angle (the acute angle to the x-axis), use its exact value, then fix the sign with ASTC.

$cos 15 0^{\circ}$ : related angle $3 0^{\circ}$ , quadrant 2 where cos is negative, so $cos 15 0^{\circ} = - \frac{3}{2}$ .

$sin \frac{4 π}{3}$ : related angle $\frac{π}{3}$ , quadrant 3 where sin is negative, so $- \frac{3}{2}$ .

Quick summary

Arc length $= \frac{θ}{360} \times 2 π r$ and sector area $= \frac{θ}{360} \times π r^{2}$ in degrees.
$π$ radians $= 18 0^{\circ}$ . In radians: $l = r θ$ and $A = \frac{1}{2} r^{2} θ$ .
On the unit circle, the point at angle $θ$ is $(cos θ, sin θ)$ .
ASTC gives the sign in each quadrant. Use the related angle for the size.
Calculus always uses radians.

Quiz

14 questions with instant marking. It is free, and your scores are saved to your account so you can carry on from any device.

Turn on JavaScript to sign in and take the quiz.