47 Angles

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 [latex]\,C=2\pi r.\,[/latex]  If we divide both sides of this equation by [latex]\,r,[/latex] we create the ratio of the circumference, which is always [latex]\,2\pi ,[/latex] to the radius, regardless of the length of the radius. So the circumference of any circle is [latex]\,2\pi \approx 6.28\,[/latex] 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).

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 [latex]\,2\pi \,[/latex] times the radius, a full circular rotation is [latex]\,2\pi \,[/latex] radians.

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.

Relating Arc Lengths to Radius

An arc length [latex]\,s\,[/latex] 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, [latex]\theta[/latex] (in radians), as the ratio of the arc length [latex]\,s\,[/latex] to the radius r. See (Figure 12).

[latex]\begin{array}{ccc}s& =& r\theta \\ \theta & =& \frac{s}{r}\end{array}[/latex]

If [latex]\,s=r,[/latex] then [latex]\,\theta =\frac{r}{r}=\text{ 1 radian}\text{.}[/latex]

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 [latex]\,C=2\pi r,[/latex] where [latex]\,r\,[/latex] is the radius. The smaller circle then has circumference [latex]\,2\pi \left(2\right)=4\pi \,[/latex] and the larger has circumference [latex]\,2\pi \left(3\right)=6\pi .\,[/latex] Now we draw a [latex]\,45^\circ\,[/latex] angle on the two circles, as in (Figure 13).

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

[latex]\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}[/latex]

Since both ratios are [latex]\,\frac{1}{4}\pi ,[/latex] 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 [latex]\,\left(360^\circ\right)\,[/latex] equals [latex]\,2\pi \,[/latex] radians. A half revolution [latex]\,\left(180^\circ\right)\,[/latex] is equivalent to [latex]\,\pi \,[/latex] 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 [latex]\,s\,[/latex] is the length of an arc of a circle, and [latex]\,r\,[/latex] is the radius of the circle, then the central angle containing that arc measures [latex]\,\frac{s}{r}\,[/latex] 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 [latex]\,60^\circ.\,[/latex] Is that correct?

Yes. It is approximately [latex]\,57.3^\circ.\,[/latex] Because [latex]\,2\pi \,[/latex] radians equals [latex]360^\circ, 1[/latex] radian equals [latex]\,\frac{360^\circ}{2\pi }\approx 57.3^\circ.[/latex]

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.

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.

Finding a Radian Measure

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

[reveal-answer q=”fs-id1726036″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1726036″]

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

[latex]1\text{ rotation}=2\pi r[/latex]

So,

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

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

[latex]\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}[/latex][/hidden-answer]

Try It

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

[reveal-answer q=”1363818″]Show Solution[/reveal-answer][hidden-answer a=”1363818″]

[latex]\frac{3\pi }{2}[/latex]

[/hidden-answer]

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 [latex]\,0^\circ\,[/latex] to [latex]\,360^\circ,[/latex] or [latex]\,0\,[/latex] to [latex]\,2\pi .\,[/latex] 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 16). The angle of [latex]\,140^\circ\,[/latex] is a positive angle, measured counterclockwise. The angle of [latex]\,–220^\circ\,[/latex] 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 [latex]\,360^\circ\,[/latex] or less than [latex]\,0^\circ\,[/latex] is coterminal with an angle between [latex]\,0^\circ\,[/latex] and [latex]\,360^\circ,[/latex] and it is often more convenient to find the coterminal angle within the range of [latex]\,0^\circ\,[/latex] to [latex]\,360^\circ\,[/latex] than to work with an angle that is outside that range.

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

An angle’s reference angle is the measure of the smallest, positive, acute angle [latex]\,t\,[/latex] formed by the terminal side of the angle [latex]\,t\,[/latex] 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 17) for examples of reference angles for angles in different quadrants.

Finding an Angle Coterminal with an Angle of Measure Greater Than [latex]\,360^\circ[/latex]

Find an angle [latex]\,\theta \,[/latex] that is coterminal with an angle measuring [latex]\,800^\circ,[/latex] where [latex]\,0^\circ\le \theta <360^\circ.[/latex]

[reveal-answer q=”fs-id1549703″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1549703″]

An angle with measure [latex]\,800^\circ\,[/latex] is coterminal with an angle with measure [latex]\,800^\circ-360^\circ=440^\circ,[/latex] but [latex]\,440^\circ\,[/latex] is still greater than [latex]\,360^\circ,[/latex] so we subtract [latex]\,360^\circ\,[/latex] again to find another coterminal angle: [latex]\,440^\circ-360^\circ=80^\circ.[/latex]

The angle [latex]\,\theta =80^\circ\,[/latex] is coterminal with [latex]\,800^\circ.\,[/latex] To put it another way, [latex]\,800^\circ\,[/latex] equals [latex]\, 80^\circ\,[/latex] plus two full rotations, as shown in (Figure 18).

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Finding an Angle Coterminal with an Angle Measuring Less Than [latex]\,0^\circ[/latex]

Show the angle with measure [latex]\,-45^\circ\,[/latex] on a circle and find a positive coterminal angle [latex]\,\alpha \,[/latex] such that [latex]\,0^\circ\le \alpha <360^\circ.[/latex]

[reveal-answer q=”fs-id1615118″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1615118″]

Since [latex]\,45^\circ\,[/latex] is half of [latex]\,90^\circ,[/latex] we can start at the positive horizontal axis and measure clockwise half of a [latex]\,90^\circ\,[/latex] angle.

Because we can find coterminal angles by adding or subtracting a full rotation of [latex]\,360^\circ,[/latex] we can find a positive coterminal angle here by adding [latex]\,360^\circ.[/latex]

[latex]-45^\circ+360^\circ=315^\circ[/latex]

We can then show the angle on a circle, as in (Figure 19).

[/hidden-answer]

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 [latex]\,2\pi ,[/latex] find a coterminal angle between 0 and [latex]\,2\pi .[/latex]

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

Finding Coterminal Angles Using Radians

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

[reveal-answer q=”2054046″]Show Solution[/reveal-answer][hidden-answer a=”2054046″]

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 [latex]\,2\pi \,[/latex] radians:

[latex]\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}[/latex]

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

[latex]\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}[/latex]

The angle [latex]\,\frac{3\pi }{4}\,[/latex] is coterminal with [latex]\,\frac{19\pi }{4},[/latex] as shown in (Figure 20).

[/hidden-answer]

Try It

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

[reveal-answer q=”fs-id1578125″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1578125″]

[latex]\,\frac{7\pi }{6}\,[/latex]

[/hidden-answer]

Determining the Length of an Arc

Recall that the radian measure [latex]\,\theta \,[/latex] of an angle was defined as the ratio of the arc length [latex]\,s\,[/latex] of a circular arc to the radius [latex]\,r\,[/latex] of the circle, [latex]\,\theta =\frac{s}{r}.\,[/latex] 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 [latex]\,s\,[/latex] subtended by an angle with measure [latex]\,\theta \,[/latex] in radians, shown in (Figure 21), is

[latex]s=r\theta[/latex]

Finding the Area of a Sector of a Circle

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 [latex]\,r\,[/latex] can be found using the formula [latex]\,A=\pi {r}^{2}.\,[/latex] If the two radii form an angle of [latex]\,\theta ,[/latex] measured in radians, then [latex]\,\frac{\theta }{2\pi }\,[/latex] 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 [latex]\,\frac{\theta }{2\pi }\,[/latex] multiplied by the entire area. (Always remember that this formula only applies if [latex]\,\theta \,[/latex] is in radians.)

Area of a Sector

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

[latex]A=\frac{1}{2}\theta {r}^{2}[/latex]

See (Figure 22).

Finding the Area of a Sector

An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in (Figure 23). What is the area of the sector of grass the sprinkler waters?

[reveal-answer q=”fs-id1708496″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1708496″]

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:

[latex]\begin{array}{ccc}\hfill 30\text{ degrees}& =& 30\cdot \frac{\pi }{180}\hfill \\ & =& \frac{\pi }{6}\text{ radians}\hfill \end{array}[/latex]

The area of the sector is then

[latex]\begin{array}{ccc}\hfill \text{Area}& =& \frac{1}{2}\left(\frac{\pi }{6}\right){\left(20\right)}^{2}\hfill \\ & \approx & 104.72\hfill \end{array}[/latex]

So the area is about [latex]\,104.72{\text{ ft}}^{2}.[/latex][/hidden-answer]

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 [latex]\,10\pi \,[/latex] inches, every second. So the linear speed of the point is [latex]\,10\pi \,[/latex]in./s. The equation for linear speed is as follows where [latex]\,v\,[/latex] is linear speed, [latex]\,s\,[/latex] is displacement, and [latex]\,t\,[/latex] is time.

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[latex]\,\frac{360\text{ degrees}}{4\text{ seconds}}=[/latex]90 degrees per second.  (Note:  1 revolution or full rotation is [latex]360^\circ[/latex] or [latex]2\pi[/latex] radians.) 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 [latex]\,\omega \,[/latex] (read as omega) is angular speed, [latex]\,\theta \,[/latex] is the angle traversed, and [latex]\,t\,[/latex] is time.

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

Substituting this into the linear speed equation gives:

Finding Angular Speed

A water wheel, shown in (Figure 24), completes 1 rotation every 5 seconds. Find the angular speed in radians per second.

[reveal-answer q=”fs-id2595818″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id2595818″]

The wheel completes 1 rotation, or passes through an angle of [latex]\,2\pi \,[/latex] radians in 5 seconds, so the angular speed would be [latex]\,\omega =\frac{2\pi }{5}\approx 1.257\,[/latex] radians per second.

[/hidden-answer]