7.1 Angles – do todo’s

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7.1 Angles – do todo’s

Learning Objectives

After completing this section, you should be able to…

Draw angles in standard position.
Convert between degrees and radians.

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 in Figure 1 can be named as ray EF, or in symbol form $\overrightarrow{EF}$ .

Illustration of Ray EF, with point F and endpoint E. — Figure 1

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 2 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 $\angle DEF.$

Illustration of Angle DEF, with vertex E and points D and F. — Figure 2

Greek letters are often used as variables for the measure of an angle. Table 1 is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 3.

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

Illustration of angle theta. — Figure 3 Angle theta, shown as $\angle\theta$

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 4.

Illustration of an angle with labels for initial side, terminal side, and vertex. — Figure 4

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 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 is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See Figure 5.

Graph of an angle in standard position with labels for the initial side and terminal side. The initial side starts on the x-axis and the terminal side is in Quadrant II with a counterclockwise arrow connecting the two. — Figure 5

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 6.

Side by side graphs. Graph on the left is a 90 degree angle and graph on the right is a 360 degree angle. Terminal side and initial side are labeled for both graphs. — Figure 6

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 Figure 7.

Four side by side graphs. First graph shows angle of 0 degrees. Second graph shows an angle of 90 degrees. Third graph shows an angle of 180 degrees. Fourth graph shows an angle of 270 degrees. — Figure 7 Quadrantal angles have a terminal side that lies along an axis. Examples are shown.

Quadrantal Angles

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

How To

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

Express the angle measure as a fraction of $\text{360°}.$
Reduce the fraction to simplest form.
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

ⓐSketch an angle of $30°$ in standard position.
ⓑSketch an angle of $−135°$ in standard position.

Solution

ⓐDivide the angle measure by $360°.$ [latex]\frac{30°}{360°}=\frac{1}{12}[/latex]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 8.

Figure 8
ⓑDivide the angle measure by $360°.$ [latex]\frac{−135°}{360°}=−\frac{3}{8}[/latex]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 in Figure 9.

Figure 9

Try It 1

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

Show Solution

Figure 1 Graph of a 240-degree angle with a counterclockwise rotation.

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 10.

Illustration of a circle showing the number of radians in a circle. A circle with points on it and between two points in counterclockwise rotation is a number which represents how many radians in that arc. — Figure 10

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.

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

See Figure 11. 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.

Illustration of a circle with angle t, radius r, and an arc of r. The — Figure 11 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 $s$ to the radius r. See Figure 12.

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

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

Three side-by-side graphs of circles. First graph has a circle with radius r and arc s, with equivalence between r and s. The second graph shows a circle with radius r and an arc of length 2r. The third graph shows a circle with a full revolution, showing 6.28 radians. — Figure 12 (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 13.

Graph of a circle with a 45-degree angle and a label for pi/4 radians. — Figure 13 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.

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

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.

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 $\frac{s}{r}$ radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.

Q&A

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 $\frac{360°}{2\pi} \approx 57.3°.$

Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 12, 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.

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

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 14. Memorizing these angles will be very useful as we study the properties associated with angles.

A graph of a circle with angles of 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees. — Figure 14 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 14, which are shown in Figure 15. Be sure you can verify each of these measures.

A graph of a circle with angles of 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees. The graph also shows the equivalent amount of radians for each angle of degrees. For example, 30 degrees is equal to pi/6 radians. — Figure 15 Commonly encountered angles measured in radians

Example 2

Finding a Radian Measure

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

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

So,

$\begin{align*} s &= \frac{1}{3}(2\pi r) \\ &= \frac{2\pi r}{3} \end{align*}$

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

$\begin{align*} \text{radian measure} &= \frac{\frac{2\pi r}{3}}{r} \\ &= \frac{2\pi r}{3r} \\ &= \frac{2\pi}{3} \end{align*}$

Try It 2

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

Show Solution

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

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.

$\frac{\pi }{6}$
3

Solution

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

We use the proportion, substituting the given information.
$\begin{align*} \frac{\theta}{180} &= \frac{\theta_R}{\pi} \\ \frac{\theta}{180} &= \frac{\frac{\pi}{6}}{\pi} \\ \theta &= \frac{180}{6} \\ \theta &= 30° \end{align*}$
We use the proportion, substituting the given information.
$\begin{align*} \frac{\theta}{180} &= \frac{\theta_{R}}{\pi} \\ \frac{\theta}{180} &= \frac{3}{\pi} \\ \theta &= \frac{3(180)}{\pi} \\ \theta &\approx 172° \end{align*}$

Try It 3

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

Show Solution

$-135^{\circ}$

Example 4

Converting Degrees to Radians

Convert $15$ degrees to radians.

Solution

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.

$\begin{align*} \frac{\theta}{180} &= \frac{\theta_R}{\pi} \\ \frac{15}{180} &= \frac{\theta_R}{\pi} \\ \frac{15\pi}{180} &= \theta_R \\ \frac{\pi}{12} &= \theta_R \end{align*}$

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 $\frac{\pi }{12}.$