Functions

Angles, Degrees, and Radians

Angle

An angle is created by two rays that intersect at a common endpoint. We use Greek letter $\theta$ to denote angles.

An angle that opens counterclockwise from the x-axis is positive.

An angle that opens clockwise from the x-axis is negative.

Degree and Radian

Angles can be measured in 2 ways:

A degree is a measure of the angle formed by $\dfrac{\displaystyle 1}{360}$ of one complete rotation of a circle.
A radian is a measure of the angle formed by the arc of a circle whose length is equal to the circle's radius.

THEOREM

Radian

\begin{aligned} \theta = {\dfrac{s}{r}} = {\dfrac{\text{arc length}}{\text{radius}}} \end{aligned}

How are radians and degrees related?

A radian compares arc length with radius. Keep the angle fixed and change the radius: the arc grows, but their ratio stays the same.

Angle (degrees)60Radius r2

\displaystyle |\theta|=\frac{s}{r}=\frac{2.094}{2}=1.047\ \text{rad}

\displaystyle \theta=60^\circ=1.047\ \text{rad}

Try doubling r. Then compare 180° with −180°: arc length is nonnegative, while the sign of the angle records counterclockwise or clockwise rotation.

THEOREM

Degrees and radians

\begin{aligned} & 180^\circ = \pi \,\text{radians} \\[0.65em] & 1^\circ = {\dfrac{\pi}{180}} \,\text{radians} \\[0.65em] & 1 \,\text{radian} = {\dfrac{180}{\pi}}^\circ \end{aligned}

This relationship provides us with a way to easily convert between the two measures.

Given any angle $\theta$ , what are these equivalent angles?

THEOREM

Equivalent angles

\begin{aligned} & \theta + 2k\pi \ (k \in \mathbb{Z}) \end{aligned}

QUESTION

Which angle is coterminal with

\displaystyle 30^\circ

Convert from degrees to radians.

(a) $\displaystyle 30^\circ$

(b) $\displaystyle 220^\circ$

Convert from radians to degrees.

(a) $\displaystyle \displaystyle \dfrac{\pi}{4}$

(b) $\displaystyle \displaystyle \dfrac{5\pi}{6}$

◀ Previous SectionFunctions Next Section ▶Trigonometric Functions

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Functions

Angles, Degrees, and Radians

Angle

An angle is created by two rays that intersect at a common endpoint. We use Greek letter $\theta$ to denote angles.

An angle that opens counterclockwise from the x-axis is positive.

An angle that opens clockwise from the x-axis is negative.

Degree and Radian

Angles can be measured in 2 ways:

A degree is a measure of the angle formed by $\dfrac{\displaystyle 1}{360}$ of one complete rotation of a circle.
A radian is a measure of the angle formed by the arc of a circle whose length is equal to the circle's radius.

THEOREM

Radian

\begin{aligned} \theta = {\dfrac{s}{r}} = {\dfrac{\text{arc length}}{\text{radius}}} \end{aligned}

How are radians and degrees related?

A radian compares arc length with radius. Keep the angle fixed and change the radius: the arc grows, but their ratio stays the same.

Angle (degrees)60Radius r2

\displaystyle |\theta|=\frac{s}{r}=\frac{2.094}{2}=1.047\ \text{rad}

\displaystyle \theta=60^\circ=1.047\ \text{rad}

Try doubling r. Then compare 180° with −180°: arc length is nonnegative, while the sign of the angle records counterclockwise or clockwise rotation.

THEOREM

Degrees and radians

\begin{aligned} & 180^\circ = \pi \,\text{radians} \\[0.65em] & 1^\circ = {\dfrac{\pi}{180}} \,\text{radians} \\[0.65em] & 1 \,\text{radian} = {\dfrac{180}{\pi}}^\circ \end{aligned}

This relationship provides us with a way to easily convert between the two measures.

Given any angle $\theta$ , what are these equivalent angles?

THEOREM

Equivalent angles

\begin{aligned} & \theta + 2k\pi \ (k \in \mathbb{Z}) \end{aligned}

QUESTION

Which angle is coterminal with

\displaystyle 30^\circ

Convert from degrees to radians.

(a) $\displaystyle 30^\circ$

(b) $\displaystyle 220^\circ$

Convert from radians to degrees.

(a) $\displaystyle \displaystyle \dfrac{\pi}{4}$

(b) $\displaystyle \displaystyle \dfrac{5\pi}{6}$

◀ Previous SectionFunctions Next Section ▶Trigonometric Functions

Angles, Degrees, and Radians

Angle

Degree and Radian

Radian

Measure an angle with arc length

Degrees and radians

Equivalent angles

KNOWLEDGE QUIZ

Degrees to radians

Radians to degrees

Angles, Degrees, and Radians

Angle

Degree and Radian

Radian

Measure an angle with arc length

Degrees and radians

Equivalent angles

KNOWLEDGE QUIZ

Degrees to radians

Radians to degrees