AP®MathsCollege BoardPrecalculusRevision NotesTrigonometric & Polar FunctionsTrigonometric FunctionsRadian Measure & Standard Position

Radian Measure & Standard Position (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

Radian measure & standard position

What is an angle in standard position?

In the coordinate plane, an angle is in standard position when:
- The vertex of the angle is at the origin
- One ray lies along the positive $x$ -axis
  - this ray is sometimes called the initial ray (or initial side)
- The other ray is called the terminal ray (or terminal side)
The measure of the angle describes the amount of rotation from the initial side to the terminal ray
- A positive angle measure indicates rotation in the counterclockwise direction
- A negative angle measure indicates rotation in the clockwise direction

Graph with x and y axes. Positive angle in green is counterclockwise, negative angle in red is clockwise, both from the positive x-axis. — **Positive and negative angle measures in standard position**

What are coterminal angles?

Two angles in standard position are coterminal if they share the same terminal ray
- Coterminal angles differ by an integer number of full revolutions
  - In degrees: coterminal angles differ by a multiple of $360 °$
  - In radians: coterminal angles differ by a multiple of $2 π$
- E.g. angles of $45 °$ and $405 °$ are coterminal
  - because $405 ° = 45 ° + 360 °$
- E.g. angles of $60 °$ and $- 300 °$ are coterminal
  - because $- 300 ° = 60 ° - 360 °$
Every angle has infinitely many coterminal angles

What is radian measure?

The radian measure of an angle in standard position is defined as
- the ratio of the arc length subtended by the angle
  - to the radius of the circle:

$θ = \frac{arc length}{radius} = \frac{s}{r}$

For a unit circle (a circle with radius $1$ centered at the origin)
- the radian measure of the angle is simply equal to the length of the arc subtended by the angle
  - because in that case $θ = \frac{arc length}{1} = arc length$
- One full revolution corresponds to the full circumference of the unit circle, which has length $2 π$
  - Therefore one full revolution = $2 π$ radians

A unit circle with radius 1 on a coordinate plane, angle θ in radians, showing arc length θ in red. Centre at origin, labelled points and axes. — **Arc length is equal to angle measure in radians on a unit circle**

Because one full revolution is $360 °$ in degrees and $2 π$ radians
- $360 ° = 2 π radians$
This gives the conversion factor

$180 ° = π radians$

To convert from degrees to radians, multiply by $\frac{π}{180}$
- E.g. $90 ° \times \frac{π}{180} = \frac{π}{2}$ radians
To convert from radians to degrees, multiply by $\frac{180}{π}$
- E.g. $\frac{3 π}{4} \times \frac{180}{π} = 135 °$

What are the radian equivalents of important angles?

Certain angles appear frequently throughout trigonometry and their radian equivalents should be memorized
- The key angles are the multiples of $30 °$ or of $45 °$ (including $0 °$ and $360 °$ )

Degrees	Radians
$0 °$	$0$
$30 °$	$\frac{π}{6}$
$45 °$	$\frac{π}{4}$
$60 °$	$\frac{π}{3}$
$90 °$	$\frac{π}{2}$
$120 °$	$\frac{2 π}{3}$
$135 °$	$\frac{3 π}{4}$
$150 °$	$\frac{5 π}{6}$
$180 °$	$π$
$210 °$	$\frac{7 π}{6}$
$225 °$	$\frac{5 π}{4}$
$240 °$	$\frac{4 π}{3}$
$270 °$	$\frac{3 π}{2}$
$300 °$	$\frac{5 π}{3}$
$315 °$	$\frac{7 π}{4}$
$330 °$	$\frac{11 π}{6}$
$360 °$	$2 π$

A useful pattern
- the multiples of $30 °$ have denominators of $6$ ( $\frac{π}{6}, \frac{2 π}{6}, \frac{3 π}{6}$ , etc.)
  - and the multiples of $45 °$ have denominators of $4$ ( $\frac{π}{4}, \frac{2 π}{4}, \frac{3 π}{4}$ , etc.)
- Although in many cases the fraction simplifies (e.g. $\frac{2 π}{6} = \frac{π}{3}$ , $\frac{3 π}{6} = \frac{2 π}{4} = \frac{π}{2}$ , etc.)

Examiner Tips and Tricks

You should be as comfortable working with angles measured in radians as you are with angles measured in degrees.

Throughout this course (and also in calculus), radians are the standard unit for angle measurement.

Practicing converting between the two systems now will save time later, and memorizing the key angle equivalents will help you work more efficiently on the exam.

Worked Example

An angle of $\frac{5 π}{3}$ radians is in standard position.

(a) Convert $\frac{5 π}{3}$ radians to degrees.

Answer:

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

$\frac{5 π}{3} \times \frac{180}{π} = \frac{5 \times 180}{3} = \frac{900}{3}$

$300 °$

(b) Find the measure, in radians, of a negative angle that is coterminal with $\frac{5 π}{3}$ .

Answer:

Coterminal angles differ by a multiple of $2 π$

So find a negative coterminal angle, subtract $2 π$

$\frac{5 π}{3} - 2 π = \frac{5 π}{3} - \frac{6 π}{3} = - \frac{π}{3}$

A negative coterminal angle is $- \frac{π}{3}$ radians
(or $- \frac{7 π}{3}$ radians, or $- \frac{13 π}{3}$ radians, etc.)

(c) A circle centered at the origin has a radius of $4$ . Find the length of the arc on this circle that is subtended by the angle $\frac{5 π}{3}$ .

Answer:

Rearrange the relationship $θ = \frac{s}{r}$ to find the arc length:

$s = r θ = 4 \times \frac{5 π}{3} = \frac{20 π}{3}$

The arc length is $\frac{20 π}{3}$ ( $\approx 20.944$ )

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Radian Measure & Standard Position (College Board AP® Precalculus): Revision Note

Radian measure & standard position

What is an angle in standard position?

What are coterminal angles?

What is radian measure?

What are the radian equivalents of important angles?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis

Radian Measure & Standard Position (College Board AP® Precalculus): Revision Note

Radian measure & standard position

What is an angle in standard position?

What are coterminal angles?

What is radian measure?

How are degrees and radians related?

What are the radian equivalents of important angles?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis