4.1.3: Angles on the Coordinate Plane

Learning Objectives

• Draw angles in standard position.
• Convert between degrees and radians.
• Find coterminal angles.
• Find the length of a circular arc.
• Use linear and angular speed to describe motion on a circular path.

Be Prepared

Before you get started, take this readiness quiz.

1. Find an integer $n$ so that   $900+n360$ is between $0$ an $360$.
2. Draw an angle measuring $300^{\text{o}}$
3. Reduce $\dfrac{270}{180}$.

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.

Angles that are larger than $90^{\text{o}}$ occur in applications and we would like to extend our ideas to deal with these angles.  For example, we would like to be able to relate lengths and angles of triangles that are not right triangles and may have an obtuse interior angle.  Here we will extend our trigonometric ratios to angles that are not interior angles of a right triangle.

Drawing Angles in Standard Position

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 $\PageIndex{1}$.

Figure $\PageIndex{1}$

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 $\dfrac{90°}{360°}=\dfrac{1}{4}$. 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 $\dfrac{360°}{360°}=1$. 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 $\PageIndex{2}$ .

Figure $\PageIndex{2}$

 

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

1. Express the angle measure as a fraction of 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 $\PageIndex{1}$ 

1. Sketch an angle of 30° in standard position.

2. Sketch an angle of −135° in standard position.

Solution

1. Divide the angle measure by 360°.

$$\dfrac{30°}{360°}=\dfrac{1}{12}\nonumber$$

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

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

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 $\PageIndex{3}$.

Figure $\PageIndex{3}$

2. Divide the angle measure by 360°.

$$\dfrac{−135°}{360°}=−\dfrac{3}{8}\nonumber$$

In this case, we can recognize that

$$−\dfrac{3}{8}=−\dfrac{3}{2}\left(\dfrac{1}{4}\right)\nonumber$$

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 $\PageIndex{4}$.

 

Figure $\PageIndex{4}$

Try It $\PageIndex{5}$

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

Answer

Figure $\PageIndex{1}$: Copy and Paste Caption here. (Copyright; author via source)

 

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 to the radius, which is always $2\pi$ 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 $\PageIndex{6}$.

Figure $\PageIndex{6}$

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. So

$$\begin{align} 2\pi \text{ radians } & =360^{\text{o}} \\ \pi \text{ radians } & =\dfrac{360^{\text{o}}}{2}=180^{\text{o}} \\ 1 \text{ radian } & =\dfrac{180^{\text{o}}}{\pi }\approx 57.3^{\text{o}} \end{align}$$

See Figure $\PageIndex{7}$. 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 out.

Figure $\PageIndex{7}$: The angle $t$ sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.

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 s to the radius $r$. See Figure $\PageIndex{8}$.

$$\begin{align} s &=r\theta \\ \theta &=\dfrac{s}{r} \end{align}$$

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

Figure $\PageIndex{8}$: (a) In an angle of 1 radian, the arc length $s$ equals the radius $r.$ (b) An angle of 2 radians has an arc length $s=2r$. (c) A full revolution is $2\pi$ or about 6.28 radians.

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 $2\pi (2)=4\pi$ and the larger has circumference $2\pi (3)=6\pi$. Now we draw a 45° angle on the two circles, as in Figure $\PageIndex{9}$.

Figure $\PageIndex{9}$: A 45° angle contains one-eighth of the circumference of a circle, regardless of the radius.

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

$$\begin{align} \text{Smaller circle: }\quad \dfrac{\dfrac{1}{2}\pi}{2} & =\dfrac{1}{4}\pi \\[2mm] \text{Larger circle: } \quad\dfrac{\dfrac{3}{4}\pi}{3} & =\dfrac{1}{4}\pi \end{align}$$

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

RADIANS

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 (360°) equals $2\pi$ radians. A half revolution (180°) 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 $\dfrac{s}{r}$ 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 $\dfrac{360°}{2\pi }\approx57.3°$.

Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure $\PageIndex{9}$, 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.

$$\begin{array}{cll} 1 \text{ rotation } & =360° & =2\pi \text{ radians} \nonumber \\[2mm] \dfrac{1}{2} \text{ rotation } & =180° & =\pi \text{ radians} \nonumber \\[2mm] \dfrac{1}{4} \text{ rotation } & =90° & =\dfrac{\pi}{2} \text{ radians} \nonumber\end{array}$$

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 $\PageIndex{10}$. 

Figure $\PageIndex{10}$: Commonly encountered angles measured in degrees

Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure $\PageIndex{10}$, which are shown in Figure $\PageIndex{11}$. Be sure you can verify each of these measures.

Figure $\PageIndex{11}$: Commonly encountered angles measured in radians

Example $\PageIndex{12}$

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

Solution

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

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

So,

$$\begin{align*}s &= \dfrac{1}{3}(2\pi r) \\[2mm] &=\dfrac{2\pi r}{3} \end{align*}$$

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

$$\begin{align*} \text{radian measure} & = \dfrac{\dfrac{2\pi r}{3}}{r} \\[2mm] &= \dfrac{2\pi r}{3r} \\[2mm] & = \dfrac{2\pi }{3} \end{align*}$$

Try It $\PageIndex{13}$

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

Solution

$\dfrac{3\pi }{2}$

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. We will use $\theta$ to represent the measure in degrees of an angle and $x$ the measurement in radians of the same angle. 

$$\dfrac{\theta}{180}=\dfrac{x}{\pi}.$$

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

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

CONVERTING BETWEEN RADIANS AND DEGREES

To convert between degrees and radians, use the proportion

$$\dfrac{\theta}{180}=\dfrac{x}{\pi}$$

Example $\PageIndex{14}$: Converting Radians to Degrees

Convert each radian measure to degrees.

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

Solution

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.

   $$\begin{align*}\dfrac{\theta}{180} &=\dfrac{x}{\pi } \\[2mm] \dfrac{\theta}{180} & =\dfrac{\dfrac{\pi }{6}}{\pi } \\ \theta & =\dfrac{180}{6} \\ \theta & =30^{\text{o}} \end{align*}$$

2. We use the proportion, substituting the given information.

   $$\begin{align*}\dfrac{\theta}{180} &= \dfrac{x}{\pi } \\ \dfrac{\theta}{180} &=\dfrac{3}{\pi } \\ \theta &= \dfrac{3(180)}{\pi } \\ \theta & \approx 172° \end{align*}$$

Try It $\PageIndex{15}$

Convert $−\dfrac{3\pi }{4}$ radians to degrees.

Solution

−135°

Example $\PageIndex{16}$

Convert $15$ degrees to radians.

Solution

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

$$\begin{align*}\dfrac{\theta}{180} &= \dfrac{x}{\pi } \\ \dfrac{15}{180} &=\dfrac{x}{\pi } \\ \dfrac{15\pi }{180} & =x \\ \dfrac{\pi }{12} & =x \end{align*}$$

Analysis

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

Try It $\PageIndex{17}$

Convert 126° to radians.

Solution

$\dfrac{7\pi}{10}$

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 $0^\text{o}$ to $360^\text{o}$, 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 $\PageIndex{18}$. The angle of $140^\text{o}$ is a positive angle, measured counterclockwise. The angle of $–220^\text{o}$ 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 $360^\text{o}$ or less than $0^\text{o}$ is coterminal with an angle between $0^\text{o}$ and $360^\text{o}$, and it is often more convenient to find the coterminal angle within the range of $0^\text{o}$ to $360^\text{o}$ than to work with an angle that is outside that range.

Figure $\PageIndex{18}$: An angle of $140^\text{o}$ and an angle of $–220^\text{o}$ are coterminal angles.

Any angle has infinitely many coterminal angles because each time we add $360^\text{o}$ to that angle—or subtract $360^\text{o}$ from it—the resulting value has a terminal side in the same location. For example, $100^\text{o}$ and $460^\text{o}$ are coterminal for this reason, as is $−260^\text{o}$. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.

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 $\PageIndex{19}$ for examples of reference angles for angles in different quadrants.

Figure $\PageIndex{19}$

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, $t′$, formed by the terminal side of the angle $t$ and the horizontal axis.

Given an angle greater than $360^\text{o}$, find a coterminal angle between $0^\text{o}$ and $360^\text{o}$

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

Example $\PageIndex{20}$

Find the least positive angle $\theta$ that is coterminal with an angle measuring $800^\text{o}$, where $0^\text{o}≤\theta<360^\text{o}$.

Solution

An angle with measure $800^\text{o}$ is coterminal with an angle with measure $800^\text{o} − 360^\text{o} = 440^\text{o}$, but $440^\text{o}$ is still greater than $360^\text{o}$, so we subtract $360^\text{o}$ again to find another coterminal angle: $440^\text{o} − 360^\text{o} = 80^\text{o}$.

The angle $\theta=80^\text{o}$ is coterminal with $800^\text{o}$. To put it another way, $800^\text{o}$ equals $80^\text{o}$ plus two full rotations, as shown in Figure $\PageIndex{21}$.

Figure $\PageIndex{21}$

Try It $\PageIndex{22}$

Find an angle $\alpha$ that is coterminal with an angle measuring $870^\text{o}$, where $0^\text{o}\leq \alpha<360^\text{o}$.

Solution

$\alpha=150^\text{o}$

Given an angle with measure less than $0^\text{o}$, find a coterminal angle having a measure between $0^\text{o}$ and $360^\text{o}$.

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

Example $\PageIndex{23}$

Show the angle with measure $-45^\text{o}$ on a circle and find a positive coterminal angle α such that $0^\text{o} \leq \alpha < 360^\text{o}$.

Solution

Since $45^\text{o}$ is half of $90^\text{o}$, we can start at the positive horizontal axis and measure clockwise half of a $90^\text{o}$  angle.

Because we can find coterminal angles by adding or subtracting a full rotation of $360^\text{o}$, we can find a positive coterminal angle here by adding $360^\text{o}$:

$$−45^\text{o}+360^\text{o}=315^\text{o}$$

We can then show the angle on a circle, as 

Try It $\PageIndex{24}$

Find an angle $\beta$ that is coterminal with an angle measuring $-300^\text{o}$ such that $0^\text{o}\leq \beta<360^\text{o}$.

Solution

$\beta=60^\text{o}$

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.

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 $\PageIndex{25}$

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

Solution

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:

$$\begin{align} \dfrac{19\pi}{4}−2\pi & =\dfrac{19\pi}{4}−\dfrac{8\pi}{4} \\ &=\dfrac{11\pi}{4} \end{align}$$

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

$$\begin{align} \dfrac{11\pi}{4}−2\pi &= \dfrac{11\pi}{4}−\dfrac{8\pi}{4} \\ &= \dfrac{3\pi}{4} \end{align}$$

The angle $\dfrac{3\pi}{4}$ is coterminal with $\dfrac{19\pi}{4}$, as shown

Try It $\PageIndex{26}$

Find an angle of measure $\theta$ that is coterminal with an angle of measure $−\dfrac{17\pi}{6}$ where $0\leq \theta<2\pi .$

Solution

$\dfrac{7\pi}{6}$

Determining the Length of an Arc (optional)

Recall that the radian measure $\theta$ of an angle was defined as the ratio of the arc length $s$ of a circular arc to the radius $r$ of the circle, $\theta=\dfrac{s}{r}$. From this relationship, we can find arc length along a circle, given an angle.

ARC LENGTH ON A CIRCLE

In a circle of radius r, the length of an arc $s$ subtended by an angle with measure $\theta$ in radians, shown in Figure $\PageIndex{27}$, is

$$s=r\theta$$

Figure $\PageIndex{27}$

Given a circle of radius $r,$ calculate the length $s$ of the arc subtended by a given angle of measure $\theta$.

1. If necessary, convert $\theta$ to radians.
2. Multiply the radius $r$ by the radian measure of $\theta$: $s=r\theta$.

Example $\PageIndex{28}$: Finding the Length of an Arc

Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

1. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
2. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.

Solution

1. Let’s begin by finding the circumference of Mercury’s orbit.

   $$\begin{align} C &=2\pi r \\ &=2\pi (\text{ 36 million miles}) \\ &\approx 226 \text{ million miles} \end{align}$$

   Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:

   $$(0.0114)226 \text{ million miles} = 2.58 \text{ million miles}$$

2. Now, we convert to radians:

   $$\begin{align} \text{radian} & = \dfrac{\text{arc length}}{\text{radius}} \\ &= \dfrac{2.58 \text{ million miles}}{36 \text{ million miles}} \\ & =0.0717 \end{align}$$

Try It $\PageIndex{29}$

Find the arc length along a circle of radius 10 units subtended by an angle of 215°.

Solution

$$\dfrac{215\pi}{18}=37.525 \text{ units}$$

Finding the Area of a Sector of a Circle (optional)

In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius $r$ can be found using the formula $A=\pi r^2$. If the two radii form an angle of $\theta$, measured in radians, then $\dfrac{\theta}{2\pi }$ is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction $\dfrac{\theta}{2\pi }$ multiplied by the entire area. (Always remember that this formula only applies if $\theta$ is in radians.)

$$\begin{align} \text{Area of sector} & =\left(\dfrac{\theta}{2\pi}\right)\pi r^2 \\ &=\dfrac{\theta\pi r^2}{2\pi } \\ & =\dfrac{1}{2}\theta r^2 \end{align}$$

AREA OF A SECTOR

The area of a sector of a circle with radius $r$ subtended by an angle $\theta$, measured in radians, is

$$A=\dfrac{1}{2}\theta r^2$$

The area of the sector equals half the square of the radius times the central angle measured in radians.

Given a circle of radius $r,$ find the area of a sector defined by a given angle $\theta.$

1. If necessary, convert $\theta$ to radians.
2. Multiply half the radian measure of $\theta$ by the square of the radius $r: A=\dfrac{1}{2}\theta r^2.$

Example $\PageIndex{30}$

An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in the figure below. What is the area of the sector of grass the sprinkler waters?

The sprinkler sprays 20 ft within an arc of 30°.

Solution

First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:

$$\begin{align} \text{30 degrees} &=30⋅\dfrac{\pi }{180} \\ &=\dfrac{\pi }{6} \text{ radians} \end{align}$$

The area of the sector is then

$$\begin{align} \text{Area} &= \dfrac{1}{2}(\dfrac{\pi }{6})(20)^2 \\ & \approx 104.72 \end{align}$$

So the area is about $\mathrm{104.72 \; ft^2}$.

Try It $\PageIndex{31}$

In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.

Solution

1.88

Use Linear and Angular Speed to Describe Motion on a Circular Path (optional)

In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10\pi inches, every second. So the linear speed of the point is $10\pi$ in./s. The equation for linear speed is as follows where $v$ is linear speed, $s$ is displacement, and $t$ is time.

$$v=\dfrac{s}{t}$$

Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as $\dfrac{360 \text{ degrees}}{4 \text{ seconds}}=$ 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where $\omega$ (read as omega) is angular speed, $\theta$ is the angle traversed, and $t$ is time.

$$\omega=\dfrac{\theta}{t}$$

Combining the definition of angular speed with the arc length equation, $s=r\theta$, we can find a relationship between angular and linear speeds. The angular speed equation can be solved for $\theta$, giving $\theta=\omega t.$ Substituting this into the arc length equation gives:

$$\begin{align}s &=r\theta \\ &=r\omega t \end{align}$$

Substituting this into the linear speed equation gives:

$$\begin{align} v & = \dfrac{s}{t} &=\dfrac{r\omega t}{t} &=r\omega \end{align}$$

ANGULAR AND LINEAR SPEED

As a point moves along a circle of radius $r,$ its angular speed, $\omega$, is the angular rotation $\theta$ per unit time, $t$.

$$\omega=\dfrac{\theta}{t}$$

The linear speed. $v$, of the point can be found as the distance traveled, arc length $s$, per unit time, $t.$

$$v=\dfrac{s}{t}$$

When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation

$$v=r\omega$$

This equation states that the angular speed in radians, $\omega$, representing the amount of rotation occurring in a unit of time, can be multiplied by the radius $r$ to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.

Given the amount of angle rotation and the time elapsed, calculate the angular speed

1. If necessary, convert the angle measure to radians.
2. Divide the angle in radians by the number of time units elapsed: $\omega=\dfrac{\theta}{t}.$
3. The resulting speed will be in radians per time unit.

Example $\PageIndex{32}$: Finding Angular Speed

A water wheel, shown in the figure, completes 1 rotation every 5 seconds. Find the angular speed in radians per second.

Solution

The wheel completes 1 rotation, or passes through an angle of $2\pi$ radians in 5 seconds, so the angular speed would be $\omega=\dfrac{2\pi }{5}\approx1.257$ radians per second.

Try It $\PageIndex{33}$

An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.

Solution

$−\dfrac{3\pi }{2}$ rad/s

Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed

1. Convert the total rotation to radians if necessary.
2. Divide the total rotation in radians by the elapsed time to find the angular speed: apply $\omega=\dfrac{\theta}{t}$.
3. Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply $v=r\omega$.

Example $\PageIndex{34}$: Finding a Linear Speed

A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.

Solution

Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.

We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:

$$\mathrm{180 \dfrac{\cancel{rotations}}{minute}⋅\dfrac{2\pi \; radians}{\cancel{rotation}}=360\pi \dfrac{radians}{minute}}$$

Using the formula from above along with the radius of the wheels, we can find the linear speed:

$$\begin{align} v & =(14 \text{ inches})(360\pi \dfrac{\text{radians}}{\text{minute}}) \\ &=5040\pi \dfrac{\text{inches}}{\text{minute}} \end{align}$$

Remember that radians are a unitless measure, so it is not necessary to include them.unitless measure, so it is not necessary to include them.

Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.

$$\mathrm{5040\pi \dfrac{\cancel{inches}}{\cancel{minute}}⋅\dfrac{1 \; \cancel{ feet}}{12 \; \cancel{ inches}}⋅\dfrac{1 \; \cancel{ mile}}{5280 \cancel{ feet}}⋅\dfrac{60 \cancel{ minutes}}{1 \; hour}\approx14.99 \; miles \; per \; hour \; (mph)}$$

Try It $\PageIndex{35}$

A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.

Solution

1655 kilometers per hour

Written Exercises $\PageIndex{36}$

1. Give an example of how to convert radians to degrees.
2. Describe how to find an angle $x$, $[0,2\pi)$, which is coterminal with $\dfrac{19\pi}{2}$ in two different ways.  Give examples.
3. How can you find the angle $300°$ in radians without using the conversion ratio (thinking in radians directly)?

 

Exit Question

1.  Convert $80°$ to radian measure.

2.  Find an angle $\theta$, $0\leq\theta<360°$, which is co-terminal with  $-960°$. 

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. 
• In addition to degrees, the measure of an angle can be described in radians. 
• To convert between degrees and radians, use the proportion $\dfrac{\theta}{180}=\dfrac{x}{\pi }$. 
• Two angles that have the same terminal side are called coterminal angles.
• We can find coterminal angles by adding or subtracting 360° or $2\pi$. 
• Coterminal angles can be found using radians just as they are for degrees. 
• The length of a circular arc is a fraction of the circumference of the entire circle. 
• The area of sector is a fraction of the area of the entire circle. 
• An object moving in a circular path has both linear and angular speed.
• The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time. 
• The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. 

 

Contributors and Attributions

• Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.