Angles and their measure
Trigonometry starts with angles. This page explains what an angle is and how to measure it in two ways: in degrees and in radians. It then uses radians to find the length of an arc, the area of a slice of a circle, and the speed of a turning wheel.
What is an angle?
A ray is a half-line: it starts at a point and goes on forever in one direction. An angle is made by two rays that start at the same point. That point is the vertex of the angle.
In trigonometry, we think of an angle as a turn. One ray, the initial side, stays in place. The other ray starts on top of it and turns about the vertex. Where it stops is the terminal side. A curved arrow shows the direction and the amount of the turn.
Angles are often named with Greek letters, such as ${\theta}$ (theta), ${\alpha}$ (alpha), and ${\beta}$ (beta).
- An angle is positive if the turn is counterclockwise (the opposite way to a clock's hands).
- An angle is negative if the turn is clockwise.
A ray can turn as far as we like, even more than a full turn. So angles can be as large as we like, positive or negative.
Standard position
An angle is in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive ${x}$-axis. We then say the angle lies in the quadrant that holds its terminal side.
In figure (a), the angle ${135^\circ}$ lies in quadrant II. In figure (b), ${-60^\circ}$ lies in quadrant IV. If the terminal side lies on an axis, the angle is called a quadrantal angle. Examples are ${90^\circ}$, ${180^\circ}$, and ${-270^\circ}$.
Degree measure
One full turn is divided into ${360}$ equal parts. Each part is one degree, written ${1^\circ}$. So
- a full turn is ${360^\circ}$,
- a half turn (a straight angle) is ${180^\circ}$,
- a quarter turn (a right angle) is ${90^\circ}$.
An angle between ${0^\circ}$ and ${90^\circ}$ is acute. An angle between ${90^\circ}$ and ${180^\circ}$ is obtuse.
Two positive angles are complementary if they add up to ${90^\circ}$, like ${30^\circ}$ and ${60^\circ}$. They are supplementary if they add up to ${180^\circ}$, like ${30^\circ}$ and ${150^\circ}$.
Minutes and seconds
For more precise work, such as maps and surveying, a degree is split into ${60}$ minutes (${60'}$), and a minute is split into ${60}$ seconds (${60''}$). So ${1'=\left(\dfrac{1}{60}\right)^\circ}$ and ${1''=\left(\dfrac{1}{3600}\right)^\circ}$. Most calculators instead use decimal degrees, such as ${24.21^\circ}$.
Example 1:
(a) Write ${24^\circ 12' 36''}$ in decimal degrees.
(b) Write ${63.47^\circ}$ in degrees, minutes, and seconds.
Solution:
(a) Divide the minutes by ${60}$ and the seconds by ${3600}$:
$\begin{align*}&24^\circ 12' 36''\\&=24+\dfrac{12}{60}+\dfrac{36}{3600}\\&=24+0.2+0.01\\&=24.21^\circ\end{align*}$
(b) Keep the ${63^\circ}$. Turn the decimal part into minutes by multiplying by ${60}$:
${0.47\times 60=28.2}$, so we have ${28'}$ and ${0.2'}$ left over.
Turn the leftover minutes into seconds, again multiplying by ${60}$:
${0.2\times 60=12}$, so we have ${12''}$.
So ${63.47^\circ=63^\circ 28' 12''}$.
Coterminal angles
In figure (c) above, ${390^\circ}$ is one full turn plus ${30^\circ}$. Its terminal side is the same as the terminal side of ${30^\circ}$. Angles in standard position with the same terminal side are called coterminal.
Adding or subtracting a full turn does not move the terminal side. So you get every angle coterminal with ${\theta}$ by adding or subtracting multiples of ${360^\circ}$:
${\theta+360^\circ k}$, where ${k}$ is any integer
Example 2: For each angle, find a positive and a negative coterminal angle. Then name the quadrant it lies in.
(a) ${460^\circ}$
(b) ${-150^\circ}$
Solution:
(a) Subtract ${360^\circ}$:
${460^\circ-360^\circ=100^\circ}$
Subtract ${360^\circ}$ again to get a negative angle:
${100^\circ-360^\circ=-260^\circ}$
So ${100^\circ}$ and ${-260^\circ}$ are coterminal with ${460^\circ}$. Since ${100^\circ}$ is between ${90^\circ}$ and ${180^\circ}$, the angle ${460^\circ}$ lies in quadrant II.
(b) Add ${360^\circ}$ to get a positive angle:
${-150^\circ+360^\circ=210^\circ}$
Subtract ${360^\circ}$ to get another negative angle:
${-150^\circ-360^\circ=-510^\circ}$
So ${210^\circ}$ and ${-510^\circ}$ are coterminal with ${-150^\circ}$. Since ${210^\circ}$ is between ${180^\circ}$ and ${270^\circ}$, the angle ${-150^\circ}$ lies in quadrant III.
Radian measure
Degrees are easy to picture, but the number ${360}$ was a choice made long ago. Calculus and physics use a more natural unit, the radian. It measures an angle by the length of the arc it cuts from a circle.
Draw a circle of radius ${r}$ with its center at the vertex of the angle. If the arc between the two sides of the angle is exactly as long as the radius, the angle is one radian.
In general, the radian measure of an angle is the arc length divided by the radius. It tells you how many radii fit along the arc.
Changing between degrees and radians
The arc of a full circle is its circumference, ${2\pi r}$. So a full turn is ${\dfrac{2\pi r}{r}=2\pi}$ radians. A full turn is also ${360^\circ}$, so ${2\pi}$ radians ${=360^\circ}$. Halving both sides gives the key fact:
${\pi\text{ radians}=180^\circ}$
Divide both sides by ${\pi}$, or by ${180}$, to change units:
$1\text{ radian}=\left(\dfrac{180}{\pi}\right)^\circ\approx 57.3^\circ$
${1^\circ=\dfrac{\pi}{180}\approx 0.01745}$ radian
- To change degrees to radians, multiply by ${\dfrac{\pi}{180}}$.
- To change radians to degrees, multiply by ${\dfrac{180}{\pi}}$.
When an angle is written with no unit, it is in radians. For example, ${\theta=2}$ means ${2}$ radians, not ${2^\circ}$.
Example 3: Change to radians.
(a) ${30^\circ}$ (b) ${135^\circ}$ (c) ${-270^\circ}$
Solution: Multiply by ${\dfrac{\pi}{180}}$ and simplify the fraction.
(a) $30\cdot\dfrac{\pi}{180}=\dfrac{30\pi}{180}=\dfrac{\pi}{6}$
(b) $135\cdot\dfrac{\pi}{180}=\dfrac{135\pi}{180}=\dfrac{3\pi}{4}$
(c) $\begin{align*}-270\cdot\dfrac{\pi}{180}&=-\dfrac{270\pi}{180}\\&=-\dfrac{3\pi}{2}\end{align*}$
Example 4: Change to degrees.
(a) ${\dfrac{5\pi}{6}}$ (b) ${\dfrac{7\pi}{4}}$ (c) ${2}$
Solution: Multiply by ${\dfrac{180}{\pi}}$.
(a) $\dfrac{5\pi}{6}\cdot\dfrac{180}{\pi}=\dfrac{5\cdot 180}{6}=150^\circ$
(b) $\dfrac{7\pi}{4}\cdot\dfrac{180}{\pi}=\dfrac{7\cdot 180}{4}=315^\circ$
(c) $2\cdot\dfrac{180}{\pi}=\dfrac{360}{\pi}\approx 114.6^\circ$
In (a) and (b), the ${\pi}$ cancels. In (c), there is no ${\pi}$ to cancel, so the answer is a decimal.
You will use these angles again and again. It is worth learning them by heart:
| Degrees | Radians |
|---|---|
| ${0^\circ}$ | ${0}$ |
| ${30^\circ}$ | ${\dfrac{\pi}{6}}$ |
| ${45^\circ}$ | ${\dfrac{\pi}{4}}$ |
| ${60^\circ}$ | ${\dfrac{\pi}{3}}$ |
| ${90^\circ}$ | ${\dfrac{\pi}{2}}$ |
| ${120^\circ}$ | ${\dfrac{2\pi}{3}}$ |
| ${135^\circ}$ | ${\dfrac{3\pi}{4}}$ |
| ${150^\circ}$ | ${\dfrac{5\pi}{6}}$ |
| ${180^\circ}$ | ${\pi}$ |
| ${270^\circ}$ | ${\dfrac{3\pi}{2}}$ |
| ${360^\circ}$ | ${2\pi}$ |
A quick way to remember them: ${180^\circ}$ is ${\pi}$, so ${90^\circ}$ is half of ${\pi}$, ${60^\circ}$ is a third, ${45^\circ}$ is a quarter, and ${30^\circ}$ is a sixth.
Arc length
The radian measure of an angle is ${\theta=\dfrac{s}{r}}$, where ${s}$ is the arc length and ${r}$ is the radius. Multiply both sides by ${r}$ to get the length of an arc:
${s=r\theta}$
Watch out: The formula ${s=r\theta}$ works only when ${\theta}$ is in radians. If the angle is given in degrees, change it to radians first.
Example 5:
(a) A circle has radius ${10}$ cm. Find the length of the arc cut off by a central angle of ${72^\circ}$.
(b) On a circle of radius ${6}$ cm, an arc is ${15}$ cm long. Find its central angle in radians and in degrees.
Solution:
(a) First change ${72^\circ}$ to radians:
$\theta=72\cdot\dfrac{\pi}{180}=\dfrac{2\pi}{5}$
Then use ${s=r\theta}$:
$\begin{align*}s&=10\cdot\dfrac{2\pi}{5}\\&=4\pi\approx 12.57\text{ cm}\end{align*}$
(b) Solve ${s=r\theta}$ for ${\theta}$:
${\theta=\dfrac{s}{r}=\dfrac{15}{6}=2.5}$ radians
In degrees:
$2.5\cdot\dfrac{180}{\pi}\approx 143.2^\circ$
Area of a sector
A sector is a slice of a circle, like a slice of pizza. It is the region between two radii and their arc, as in the figure above.
A full circle has area ${\pi r^2}$ and an angle of ${2\pi}$ radians. A sector with angle ${\theta}$ is the fraction ${\dfrac{\theta}{2\pi}}$ of the circle. So its area is
${A=\dfrac{\theta}{2\pi}\cdot\pi r^2}$, which simplifies to
${A=\dfrac{1}{2}r^2\theta}$
Again, ${\theta}$ must be in radians.
Example 6: A lawn sprinkler sprays water ${20}$ feet. It turns back and forth through an angle of ${135^\circ}$. Find the area of lawn it waters.
Solution:
The watered region is a sector with radius ${r=20}$ ft. Change the angle to radians:
$\theta=135\cdot\dfrac{\pi}{180}=\dfrac{3\pi}{4}$
Use the sector formula:
$\begin{align*}A&=\dfrac{1}{2}(20)^2\cdot\dfrac{3\pi}{4}\\&=\dfrac{1}{2}\cdot 400\cdot\dfrac{3\pi}{4}\\&=150\pi\approx 471\text{ ft}^2\end{align*}$
Angular speed and linear speed
Think of a point on the edge of a turning wheel. We can describe its motion in two ways:
- The angular speed ${\omega}$ (omega) is how fast the angle changes. If the wheel turns through angle ${\theta}$ in time ${t}$, then ${\omega=\dfrac{\theta}{t}}$. Its units are radians per unit of time, such as radians per second.
- The linear speed ${v}$ is how fast the point moves along the circle. If it travels a distance ${s}$ in time ${t}$, then ${v=\dfrac{s}{t}}$.
Since ${s=r\theta}$, we get $v=\dfrac{r\theta}{t}=r\cdot\dfrac{\theta}{t}$. So the two speeds are linked by
${v=r\omega}$
Here ${\omega}$ must be in radians per unit of time. One revolution (one full turn) is ${2\pi}$ radians, so a speed in revolutions per minute (rpm) is changed to radians per minute by multiplying by ${2\pi}$.
Example 7: A bicycle wheel has a radius of ${13}$ inches. It turns at ${200}$ revolutions per minute.
(a) Find the angular speed of the wheel in radians per minute.
(b) Find the speed of the bicycle in inches per minute and in miles per hour.
Solution:
(a) Each revolution is ${2\pi}$ radians:
$\begin{align*}\omega&=200\cdot 2\pi\\&=400\pi\approx 1257\text{ radians per minute}\end{align*}$
(b) The bicycle moves as fast as a point on the edge of its wheel. Use ${v=r\omega}$:
$\begin{align*}v&=13\cdot 400\pi\\&=5200\pi\\&\approx 16{,}336\end{align*}$
So the bicycle moves about ${16{,}336}$ inches per minute.
To get miles per hour, multiply by ${60}$ (there are ${60}$ minutes in an hour). Then divide by ${63{,}360}$ (a mile is ${5280}$ feet, and ${5280\cdot 12=63{,}360}$ inches):
$\begin{align*}v&\approx\dfrac{16{,}336\cdot 60}{63{,}360}\\&\approx 15.5\text{ miles per hour}\end{align*}$
Summary
- An angle is a turn from the initial side to the terminal side. Counterclockwise is positive, and clockwise is negative.
- In standard position, the vertex is at the origin, and the initial side is on the positive ${x}$-axis.
- Coterminal angles share a terminal side. They differ by multiples of ${360^\circ}$, or ${2\pi}$.
- ${\pi}$ radians ${=180^\circ}$. To change degrees to radians, multiply by ${\dfrac{\pi}{180}}$. To change radians to degrees, multiply by ${\dfrac{180}{\pi}}$.
- With ${\theta}$ in radians: arc length ${s=r\theta}$, sector area ${A=\dfrac{1}{2}r^2\theta}$, and linear speed ${v=r\omega}$.