Radian Measure & Standard Position (College Board AP® Precalculus): Study Guide

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

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 $2\pi$

  • 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:

$\theta =\frac{\text{arc length}}{\text{radius}}=\frac{s}{r}$

• 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 $\theta =\frac{\text{arc length}}{1}=\text{arc length}$

  • One full revolution corresponds to the full circumference of the unit circle, which has length $2\pi$

    • Therefore one full revolution = $\mathbf{2}\mathit{\pi }$ radians

How are degrees and radians related?

• Because one full revolution is $360°$ in degrees and $2\pi$ radians

  • $360°=2\pi \text{ radians}$

• This gives the conversion factor

$\mathbf{180}\mathbf{°}\mathbf{=}\mathit{\pi }\mathbf{\text{ radians}}$

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

  • E.g. $90°\times \frac{\pi }{180}=\frac{\pi }{2}$ radians

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

  • E.g. $\frac{3\pi }{4}\times \frac{180}{\pi }=135°$

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 $360°$)

• A useful pattern

  • the multiples of $30°$ have denominators of $6$ ($\frac{\pi }{6},\frac{2\pi }{6},\frac{3\pi }{6}$, etc.)

    • and the multiples of $45°$ have denominators of $4$ ($\frac{\pi }{4},\frac{2\pi }{4},\frac{3\pi }{4}$, etc.)

  • Although in many cases the fraction simplifies (e.g. $\frac{2\pi }{6}=\frac{\pi }{3}$, $\frac{3\pi }{6}=\frac{2\pi }{4}=\frac{\pi }{2}$, etc.)