Section 4.1: Introduction to Angles

In this section you will:

• 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.

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.

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 pictured below can be named as ray EF, or in symbol form $\overrightarrow{EF}$.

The angle shown below is formed from $\overrightarrow{ED}$ and $\overrightarrow{EF}$. 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. The table below has a list of Greek letters commonly used to represent angles, and a sample angle is shown below.

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 shown below.

Angle creation

Imagine two rays starting at the same place. Leave one in place (this is the initial side) and sweep the other ray around (this is the terminal side). This creates an angle, as pictured below.

You can play around with dynamically creating an angle by following this geogebra link.

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.

Probably the most familiar unit of angle measurement is the degree.

To formalize our work, we will begin by drawing angles on an x-y 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 in standard position

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 the image below.

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 $\frac{360^\circ}{360^\circ} = 1$. 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 below.

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 0°, 90°, 180°, 270°, or 360°. See below.

Quadrantal Angles

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

Drawing an Angle in Standard Position Measured in Degrees

A. Sketch an angle of $30^\circ$ in standard position

Click for solution

Divide the angle measure by $360^\circ$: $$\frac{30^\circ}{360^\circ} = \frac{1}{12}$$ 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 the picture below:

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

Click for solution

Divide the angle measure by $360^\circ$: $$\frac{-135^\circ}{360^\circ} = -\frac{3}{8}$$ In this case, we can recognize that $$-\frac{3}{8} = -\frac{3}{2}\left(\frac{1}{4}\right)$$ 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 shown below:

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.

If we divide both sides of this circumference 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 below

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. Check out the animation below to see how a radian is formed:

So this is how we form a radian: take the radius of the circle, lay it along the circumference. Connect the endpoints of that radius-length curve to the center of the circle. The angle formed is one radian.

Radian

A radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.

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.

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. Since we know that a whole rotation of a circle is $2 \pi$ radians, it follows that half a circle is $\pi$ radians. In degrees, half a circle is $180^\circ$. So we create a proportion where we compare radians to degrees: $$\frac{\theta ^\circ}{180^\circ} = \frac{\theta_R}{\pi}$$ We can solve this proportion for either $\theta^\circ$ or $\theta_R$ to see how to convert between degrees and radians:

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 the figure below. 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, 100° and 460° are coterminal for this reason, as is −260°. You can experiment with coterminal angles using the interactive applet found through this link by Math Open Reference.

Determining the Length of an Arc

An arc of a circle is a small chunk of its circumference, as pictured below:

In this figure, we have an arc of length $s$ subtended by an angle measure of $\theta$ in a circle of radius $r$. We can compare this small portion of the circle to the whole circle by setting up a proportion. That proportion will look slightly different depending on whether we have the central angle, $\theta$, given in degrees or in radians. In general, though, we want to compare the part of the circle to the whole. In the following proportion, we put the partial arc information on the left side of the proportion, and the whole circle information on the right: $$\frac{\text{arc length}}{\text{central angle subtending the arc}} = \frac{\text{circumference}}{\text{total angle measure of circle}}$$ Symbolically, that is... $$\frac{s}{\theta} = \frac{2\pi r}{\text{total angle measure of circle}}$$ What we use for that total angle measure will depend on whether we are given $\theta$ in degrees or radians.

Finding the area of a sector

A sector is a chunk of a circle, like a pizza slice. In the image below, a sector subtended by an angle of measure $\theta$ is shaded in yellow:

The good news here is that finding the area of that pizza slice is basically the same process as finding an arc length, but now we'll use the area formula in our proportion instead of the circumference. $$\frac{\text{sector area}}{\text{central angle subtending the sector}} = \frac{\text{circle area}}{\text{total angle measure of circle}}$$ Symbolically, that is... $$\frac{a}{\theta} = \frac{\pi r^2}{\text{total angle measure of circle}}$$ What we use for that total angle measure will depend on whether we are given $\theta$ in degrees or radians.

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

Click for solution

The angle is given in degrees, and we want to find a sector area, so we use the proportion $$\frac{a}{\theta} = \frac{\pi r^2}{360^\circ}$$ Here, $\theta = 30^\circ$, $r = 20$ and $a$ is unknown. $$\frac{a}{30} = \frac{\pi(20^2)}{360}$$ Multiply both sides by 30 to isolate $a$ and evaluate. $$a = \frac{400\pi}{360}\cdot 30 \approx 104.72$$ The area is about 104.72 $ft^2$.

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

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 10π inches, every second. So the linear speed of the point is 10π in./s. Really here, we're talking about the relationship between distance, rate, and time, which you may recognize as the equation $$d = rt$$ We want to reframe this a bit, so we'll think of the rate as $v$ and the distance as $s$ (like arc length): $$s = vt$$ If we solve for the speed, $v$, we have $$v = \frac{s}{t}$$

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 $\frac{360\text{degrees}}{4\text{ seconds}} = 90$ 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, θ is the angle traversed, and t is time: $$\omega = \frac{\theta}{t}$$

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

Finding Angular Speed

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

Click for solution

The wheel completes 1 rotation, or passes through an angle of 2π radians in 5 seconds, so the angular speed would be $$\omega = \frac{2\pi}{5} \approx 1.257$$ That's about 1.257 radians per second.

Exercises

In the following exercises, convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$.

In the following exercises, convert the angle from radian measure into degree measure.

For the following exercises, find the angle between $0^\circ$ and $360^\circ$ that is coterminal to the given angle.

17. $-40^\circ$

18. $-110^\circ$

19. $700^\circ$

20. $1400^\circ$

For the following exercises, find the angle between 0 and 2π in radians that is coterminal to the given angle.

21. $-\frac{\pi}{9}$

22. $\frac{10\pi}{3}$

23. $\frac{13\pi}{6}$

24. $\frac{44\pi}{9}$

25. A yo-yo which is 2.25 inches in diameter spins at a rate of 4500 revolutions per minute. How fast is the edge of the yo-yo spinning in miles per hour? Round your answer to two decimal places.

26. How many revolutions per minute would the yo-yo in the previous exercise have to complete if the edge of the yo-yo is to be spinning at a rate of 42 miles per hour? Round your answer to two decimal places.

27. In the yo-yo trick `Around the World,' the performer throws the yo-yo so it sweeps out a vertical circle whose radius is the yo-yo string. If the yo-yo string is 28 inches long and the yo-yo takes 3 seconds to complete one revolution of the circle, compute the speed of the yo-yo in miles per hour. Round your answer to two decimal places.

28. A computer hard drive contains a circular disk with diameter 2.5 inches and spins at a rate of 7200 revolutions per minute. Find the linear speed of a point on the edge of the disk in miles per hour.

29. A rock got stuck in the tread of my tire and when I was driving 70 miles per hour, the rock came loose and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling when it came out of the tread? (The tire has a diameter of 23 inches.)

30. The Giant Wheel at Cedar Point is a circle with diameter 128 feet which sits on an 8 foot tall platform making its overall height is 136 feet. It completes two revolutions in 2 minutes and 7 seconds. Assuming the riders are at the edge of the circle, how fast are they traveling in miles per hour?

In the following exercises, compute the areas of the circular sectors with the given central angles and radii.

37. Imagine a rope tied around the Earth at the equator. Show that you need to add only $2\pi$ feet of length to the rope in order to lift it one foot above the ground around the entire equator. (You do NOT need to know the radius of the Earth to show this.)

For additional exercises, check out the bottom of the Section 7.1 at OpenStax by following this link.

Homework Set

1. Convert $80^\circ$ to radians.
2. What is the exact radian measure of a $-182^\circ$ angle?
3. Convert $\frac{3\pi}{7}$ to degrees.
4. Convert $\frac{11\pi}{18}$ to degrees.
5. Find an angle between $0^\circ$ and $360^\circ$ that is coterminal with each of the following angles.
   1. $491^\circ$
   2. $-29^\circ$
   3. $-1715^\circ$
   4. $4572^\circ$
6. Find an angle between $0$ and $2\pi$ that is coterminal with each of the following angles.
   1. $-\frac{104\pi}{11}$
   2. $-\frac{33\pi}{10}$
   3. $\frac{17\pi}{6}$
   4. $\frac{11\pi}{10}$
7. Find the length of an arc if the radius of the arc is 6 inches and the measure of the central angle is $\frac{9\pi}{8}$.
8. A Ferris wheel has a diameter of 50 feet. How far has a person on the Ferris wheel traveled when they have moved $270^\circ$?
9. Find the area of the sector of a circle with radius 6 km formed by a central angle of $\frac{\pi}{6}$.
10. A hamster runs at a speed of 7 cm/sec in a wheel with radius 10 cm.
    1. What is the angular velocity of the wheel in radians/sec?
    2. How fast will the wheel spin in revolutions per minute?
11. A toy car has wheels with 10 cm radius. The wheels are spinning at a rate of one revolution in 6 seconds.
    1. What is the angular velocity of the wheel in radians/sec?
    2. At what linear velocity is the hamster running?
12. A truck with 36 inch diameter wheels is traveling 45 mph.
    1. Find the angular speed of the wheels in rad/min.
    2. How many revolutions per minute do the wheels make?
13. A central angle has a measure of 162 degrees and the intercepted arc's sector has an area measure of 84 $cm^2$. What is the radius of the circle? Round answers to three decimal places when necessary.