### Learning Objectives

In this section you will:

7.1.3 – Find coterminal angles.

A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.

### 7.1.1 – Drawing Angles in Standard Position

Properly defining an angle first requires that we define a ray. A ray is a directed line segment. It consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in (Figure) can be named as ray EF, or in symbol form $\,\stackrel{⟶}{EF}.$

An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in (Figure) is formed from $\,\stackrel{⟶}{ED}\,$ and $\,\stackrel{⟶}{EF}\,$ . Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form $\,\angle DEF.$

Greek letters are often used as variables for the measure of an angle. (Figure) is a list of Greek letters commonly used to represent angles, and a sample angle is shown in (Figure).

$\theta$ $\phi \,$ or $\,\varphi$ $\alpha$ $\beta$ $\gamma$
theta phi alpha beta gamma

Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arrow close to the vertex as in (Figure).

As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is $\,\frac{1}{360}\,$ of a circular rotation, so a complete circular rotation contains $\,360\,$ degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol $°.\,$ For example, $\,90\text{ degrees}=90°.$

To formalize our work, we will begin by drawing angles on an xy coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See (Figure).

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.

Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by $\,360°.\,$ For example, to draw a $\,90°\,$ angle, we calculate that $\,\frac{90°}{360°}=\frac{1}{4}.\,$ So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a $\,360°$ angle, we calculate that $\,\frac{360°}{360°}=1.\,$ So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap. See (Figure).

Since we define an angle in standard position by its terminal side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of $\text{0°,}\,\text{90°,}\,\text{180°,}\,\text{270°,}$ or $\,\text{360°}.\,$ See (Figure).

An angle is a quadrantal angle if its terminal side lies on an axis, including $\text{0°,}\,\text{90°,}\,\text{180°,}\,\text{270°,}$ or $\,\text{360°}.$

### How To

Given an angle measure in degrees, draw the angle in standard position.

1. Express the angle measure as a fraction of $\,\text{360°}.$
2. Reduce the fraction to simplest form.
3. Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles.

### Example 1 – Drawing an Angle in Standard Position Measured in Degrees

1. Sketch an angle of $\,30°\,$ in standard position.
2. Sketch an angle of $\,-135°\,$ in standard position.
1. Divide the angle measure by $\,360°.$

$\frac{30°}{360°}=\frac{1}{12}$

To rewrite the fraction in a more familiar fraction, we can recognize that

$\frac{1}{12}=\frac{1}{3}\left(\frac{1}{4}\right)$

One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at $\,30°,$ as in (Figure).

2. Divide the angle measure by $\,360°.$

$\frac{-135°}{360°}=-\frac{3}{8}$

In this case, we can recognize that

$-\frac{3}{8}=-\frac{3}{2}\left(\frac{1}{4}\right)$

Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in (Figure).

### Try It

Show an angle of $\,240°\,$ on a circle in standard position.

### 7.1.2 – Converting Between Degrees and Radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.

The circumference of a circle is $\,C=2\pi r.\,$ If we divide both sides of this equation by $\,r,$ we create the ratio of the circumference, which is always $\,2\pi ,$ to the radius, regardless of the length of the radius. So the circumference of any circle is $\,2\pi \approx 6.28\,$ times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in (Figure).

This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals $\,2\pi \,$ times the radius, a full circular rotation is $\,2\pi \,$ radians.

$\begin{array}{ccc}\hfill 2\pi \text{ radians}& =& 360°\hfill \\ \hfill \pi \text{ radians}& =& \frac{360°}{2}=180°\hfill \\ \hfill 1\text{ radian}& =& \frac{180°}{\pi }\approx 57.3°\hfill \end{array}$

See (Figure). Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel.

#### Relating Arc Lengths to Radius

An arc length $\,s\,$ is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.

This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length $\,s\,$ to the radius r. See (Figure).

$\begin{array}{ccc}s& =& r\theta \\ \theta & =& \frac{s}{r}\end{array}$

If $\,s=r,$ then $\,\theta =\frac{r}{r}=\text{ 1 radian}\text{.}$

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is $\,C=2\pi r,$ where $\,r\,$ is the radius. The smaller circle then has circumference $\,2\pi \left(2\right)=4\pi \,$ and the larger has circumference $\,2\pi \left(3\right)=6\pi .\,$ Now we draw a $\,45°\,$ angle on the two circles, as in (Figure).

Notice what happens if we find the ratio of the arc length divided by the radius of the circle.

$\begin{array}{ccc}\text{Smaller circle: }\frac{\frac{1}{2}\pi }{2}& =& \frac{1}{4}\pi \\ \text{Larger circle: }\frac{\frac{3}{4}\pi }{3}& =& \frac{1}{4}\pi \end{array}$

Since both ratios are $\,\frac{1}{4}\pi ,$ the angle measures of both circles are the same, even though the arc length and radius differ.

One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution $\,\left(360°\right)\,$ equals $\,2\pi \,$ radians. A half revolution $\,\left(180°\right)\,$ is equivalent to $\,\pi \,$ radians.

The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if $\,s\,$ is the length of an arc of a circle, and $\,r\,$ is the radius of the circle, then the central angle containing that arc measures $\,\frac{s}{r}\,$ radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.

A measure of 1 radian looks to be about $\,60°.\,$ Is that correct?

Yes. It is approximately $\,57.3°.\,$ Because $\,2\pi \,$ radians equals $360°,1$ radian equals $\,\frac{360°}{2\pi }\approx 57.3°.$

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in (Figure), suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.

Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, $\,360°.$ We can also track one rotation around a circle by finding the circumference, $\,C=2\pi r,$ and for the unit circle $\,C=2\pi .\,$ These two different ways to rotate around a circle give us a way to convert from degrees to radians.

$\begin{array}{ccccc}\hfill \text{1 rotation}& =& 360°\hfill & =& 2\pi \text{ radians}\hfill \\ \hfill \frac{1}{2}\text{ rotation}& =& 180°\hfill & =& \pi \text{ radians}\hfill \\ \hfill \frac{1}{4}\text{ rotation}& =& 90°\hfill & =& \frac{\pi }{2}\text{ radians}\hfill \end{array}$

#### Identifying Special Angles Measured in Radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in (Figure). Memorizing these angles will be very useful as we study the properties associated with angles.

Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in (Figure), which are shown in (Figure). Be sure you can verify each of these measures.

### Example 2 – Finding a Radian Measure

Find the radian measure of one-third of a full rotation.

For any circle, the arc length along such a rotation would be one-third of the circumference. We know that

$1\text{ rotation}=2\pi r$

So,

$\begin{array}{ccc}\hfill s& =& \frac{1}{3}\left(2\pi r\right)\hfill \\ & =\hfill & \frac{2\pi r}{3}\hfill \end{array}$

The radian measure would be the arc length divided by the radius.

$\begin{array}{ccc}\hfill \text{radian measure}& =& \frac{\frac{2\pi r}{3}}{r}\hfill \\ & =& \frac{2\pi r}{3r}\hfill \\ & =& \frac{2\pi }{3}\end{array}$

### Try It

Find the radian measure of three-fourths of a full rotation.

$\frac{3\pi }{2}$

#### Converting Between Radians and Degrees

Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion where $\,\theta \,$ is the measure of the angle in degrees and $\,{\theta }_{R}\,$ is the measure of the angle in radians.

$\frac{\theta }{180}=\frac{{\theta }_{{}^{R}}}{\pi }$

This proportion shows that the measure of angle $\,\theta \,$ in degrees divided by 180 equals the measure of angle $\,\theta \,$ in radians divided by $\,\pi .\,$ Or, phrased another way, degrees is to 180 as radians is to $\,\pi .$

$\frac{\text{Degrees}}{180}=\frac{\text{Radians}}{\pi }$

### Converting between Radians and Degrees

To convert between degrees and radians, use the proportion

$\frac{\theta }{180}=\frac{{\theta }_{R}}{\pi }$

### Example 3 – Converting Radians to Degrees

Convert each radian measure to degrees.

1. $\frac{\pi }{6}$
2. 3

Because we are given radians and we want degrees, we should set up a proportion and solve it.

1. We use the proportion, substituting the given information.
$\begin{array}{ccc}\hfill \frac{\theta }{180}& =& \frac{{\theta }_{R}}{\pi }\hfill \\ \hfill \frac{\theta }{180}& =& \frac{\frac{\pi }{6}}{\pi }\hfill \\ \hfill \theta & =& \frac{180}{6}\hfill \\ \hfill \theta & =& 30°\hfill \end{array}$
2. We use the proportion, substituting the given information.
$\begin{array}{ccc}\hfill \frac{\theta }{180}& =& \frac{{\theta }_{{}^{R}}}{\pi }\hfill \\ \hfill \frac{\theta }{180}& =& \frac{3}{\pi }\hfill \\ \hfill \theta & =& \frac{3\left(180\right)}{\pi }\hfill \\ \hfill \theta & \approx & 172°\hfill \end{array}$

### Try It

Convert $\,-\frac{3\pi }{4}\,$ radians to degrees.

$-135°$

### Example 4 – Converting Degrees to Radians

Convert $\,15\,$ degrees to radians.

In this example, we start with degrees and want radians, so we again set up a proportion, but we substitute the given information into a different part of the proportion.

$\begin{array}{ccc}\hfill \frac{\theta }{180}& =& \frac{{\theta }_{R}}{\pi }\hfill \\ \hfill \frac{15}{180}& =& \frac{{\theta }_{R}}{\pi }\hfill \\ \hfill \frac{15\pi }{180}& =& {\theta }_{R}\hfill \\ \hfill \frac{\pi }{12}& =& {\theta }_{R}\hfill \end{array}$

#### Analysis

Another way to think about this problem is by remembering that $\,30°=\frac{\pi }{6}.\,$ Because $\,15°=\frac{1}{2}\left(30°\right),$ we can find that $\,\frac{1}{2}\left(\frac{\pi }{6}\right)\,$ is $\,\frac{\pi }{12}.$

### Try It

Convert $\,126°\,$ to radians.

$\frac{7\pi }{10}$

### 7.1.3 – Finding Coterminal Angles

Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of $\,0°\,$ to $\,360°,$ or $\,0\,$ to $\,2\pi .\,$ It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.

It is possible for more than one angle to have the same terminal side. Look at (Figure). The angle of $\,140°\,$ is a positive angle, measured counterclockwise. The angle of $\,–220°\,$ is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than $\,360°\,$ or less than $\,0°\,$ is coterminal with an angle between $\,0°\,$ and $\,360°,$ and it is often more convenient to find the coterminal angle within the range of $\,0°\,$ to $\,360°\,$ than to work with an angle that is outside that range.

Any angle has infinitely many coterminal angles because each time we add $\,360°\,$ to that angle—or subtract $\,360°\,$ from it—the resulting value has a terminal side in the same location. For example, $\,\text{100°}\,$ and $\,\text{460°}\,$ are coterminal for this reason, as is $\,-260°.\,$

An angle’s reference angle is the measure of the smallest, positive, acute angle $\,t\,$ formed by the terminal side of the angle $\,t\,$ and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See (Figure) for examples of reference angles for angles in different quadrants.

### Coterminal and Reference Angles

Coterminal angles are two angles in standard position that have the same terminal side.

An angle’s reference angle is the size of the smallest acute angle, $\,{t}^{\prime },$ formed by the terminal side of the angle $\,t\,$ and the horizontal axis.

### How To

Given an angle greater than $\,360°,$ find a coterminal angle between $\,0°\,$ and $\,360°$

1. Subtract $\,360°\,$ from the given angle.
2. If the result is still greater than $\,360°,$ subtract $\,360°\,$ again till the result is between $\,0°\,$ and $\,360°.$
3. The resulting angle is coterminal with the original angle.

### Example 5 – Finding an Angle Coterminal with an Angle of Measure Greater Than $\,360°$

Find the least positive angle $\,\theta \,$ that is coterminal with an angle measuring $\,800°,$ where $\,0°\le \theta <360°.$

An angle with measure $\,800°\,$ is coterminal with an angle with measure $\,800-360=440°,$ but $\,440°\,$ is still greater than $\,360°,$ so we subtract $\,360°\,$ again to find another coterminal angle: $\,440-360=80°.$

The angle $\,\theta =80°\,$ is coterminal with $\,800°.\,$ To put it another way, $\,800°\,$ equals $\,80°\,$ plus two full rotations, as shown in (Figure).

### Try It

Find an angle $\,\alpha \,$ that is coterminal with an angle measuring $\,870°,$ where $\,0°\le \alpha <360°.$

$\alpha =150°$

### How To

Given an angle with measure less than $\,0°,$ find a coterminal angle having a measure between $\,0°\,$ and $\,360°.$

1. Add $\,360°\,$ to the given angle.
2. If the result is still less than $\,0°,$ add $\,360°\,$ again until the result is between $\,0°\,$ and $\,360°.$
3. The resulting angle is coterminal with the original angle.

### Example 6 – Finding an Angle Coterminal with an Angle Measuring Less Than $\,0°$

Show the angle with measure $\,-45°\,$ on a circle and find a positive coterminal angle $\,\alpha \,$ such that $\,0°\le \alpha <360°.$

Since $\,45°\,$ is half of $\,90°,$ we can start at the positive horizontal axis and measure clockwise half of a $\,90°\,$ angle.

Because we can find coterminal angles by adding or subtracting a full rotation of $\,360°,$ we can find a positive coterminal angle here by adding $\,360°.$

$-45°+360°=315°$

We can then show the angle on a circle, as in (Figure).

### Try It

Find an angle $\,\beta \,$ that is coterminal with an angle measuring $\,-300°\,$ such that $\,0°\le \beta <360°.$

$\beta =60°$

#### Finding Coterminal Angles Measured in Radians

We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

### How To

Given an angle greater than $\,2\pi ,$ find a coterminal angle between 0 and $\,2\pi .$

1. Subtract $\,2\pi \,$ from the given angle.
2. If the result is still greater than $\,2\pi ,$ subtract $\,2\pi \,$ again until the result is between $\,0\,$ and $\,2\pi .$
3. The resulting angle is coterminal with the original angle.

### Example 7 – Finding Coterminal Angles Using Radians

Find an angle $\,\beta \,$ that is coterminal with $\,\frac{19\pi }{4},$ where $\,0\le \beta <2\pi .$

When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of $\,2\pi \,$ radians:

$\begin{array}{ccc}\hfill \frac{19\pi }{4}-2\pi & =& \frac{19\pi }{4}-\frac{8\pi }{4}\hfill \\ & =& \frac{11\pi }{4}\hfill \end{array}$

The angle $\,\frac{11\pi }{4}\,$ is coterminal, but not less than $\,2\pi ,$ so we subtract another rotation.

$\begin{array}{ccc}\hfill \frac{11\pi }{4}-2\pi & =& \frac{11\pi }{4}-\frac{8\pi }{4}\hfill \\ & =& \frac{3\pi }{4}\hfill \end{array}$

The angle $\,\frac{3\pi }{4}\,$ is coterminal with $\,\frac{19\pi }{4},$ as shown in (Figure).

### Try It

Find an angle of measure $\,\theta \,$ that is coterminal with an angle of measure $\,-\frac{17\pi }{6}\,$ where $\,0\le \theta <2\pi .$

$\,\frac{7\pi }{6}\,$

Access these online resources for additional instruction and practice with angles, arc length, and areas of sectors.

### Key Equations

 arc length $s=r\theta$ area of a sector $A=\frac{1}{2}\theta {r}^{2}$ angular speed $\omega =\frac{\theta }{t}$ linear speed $v=\frac{s}{t}$ linear speed related to angular speed $v=r\omega$

### Key Concepts

• An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.
• An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.
• To draw an angle in standard position, draw the initial side along the positive x-axis and then place the terminal side according to the fraction of a full rotation the angle represents. See Example 1.
• In addition to degrees, the measure of an angle can be described in radians. See Example 2.
• To convert between degrees and radians, use the proportion $\,\frac{\theta }{180}=\frac{{\theta }_{R}}{\pi }.\,$ See Example 3 and Example 4.
• Two angles that have the same terminal side are called coterminal angles.
• We can find coterminal angles by adding or subtracting $\,360°\,$ or $\,2\pi .\,$ See Example 5 and Example 6.
• Coterminal angles can be found using radians just as they are for degrees. See Example 7.

### Glossary

angle
the union of two rays having a common endpoint
angular speed
the angle through which a rotating object travels in a unit of time
arc length
the length of the curve formed by an arc
area of a sector
area of a portion of a circle bordered by two radii and the intercepted arc; the fraction $\,\frac{\theta }{2\pi }.\,$ multiplied by the area of the entire circle
coterminal angles
description of positive and negative angles in standard position sharing the same terminal side
degree
a unit of measure describing the size of an angle as one-360th of a full revolution of a circle
initial side
the side of an angle from which rotation begins
linear speed
the distance along a straight path a rotating object travels in a unit of time; determined by the arc length
measure of an angle
the amount of rotation from the initial side to the terminal side
negative angle
description of an angle measured clockwise from the positive x-axis
positive angle
description of an angle measured counterclockwise from the positive x-axis
an angle whose terminal side lies on an axis
the ratio of the arc length formed by an angle divided by the radius of the circle
the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle
ray
one point on a line and all points extending in one direction from that point; one side of an angle
reference angle
the measure of the acute angle formed by the terminal side of the angle and the horizontal axis
standard position
the position of an angle having the vertex at the origin and the initial side along the positive x-axis
terminal side
the side of an angle at which rotation ends
vertex
the common endpoint of two rays that form an angle