### Learning Outcomes

- Draw angles in standard position.
- Convert between degrees and radians.
- Find coterminal angles.
- Find the length of a circular arc.
- Find the area of a sector of a circle.
- 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.

## Draw angles in standard position

Properly defining an angle first requires that we define a ray. A**ray**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 1can be named as ray EF, or in symbolic form [latex]\overrightarrow{EF}[/latex].

**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 2is formed from[latex]\overrightarrow{ED}[/latex] and[latex]\overrightarrow{EF}[/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[/latex], or in symbol form [latex]\angle{DEF}[/latex].

**Figure 2**

Greek letters are often used as variables for the measure of an angle. The table belowis a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 2.

[latex]\theta[/latex] | [latex]\phi \text{ or }\varphi[/latex] | [latex]\alpha[/latex] | [latex]\beta[/latex] | [latex]\gamma[/latex] |

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 arc and arrow close to the vertex as in Figure 4.

The following video provides an illustration of angles in standard position.

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 [latex]\frac{1}{360}[/latex] 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 degrees = 90°.

**Figure 5**

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.

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 angl****e**.

**Figure 6**

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 [latex]\frac{90^\circ }{360^\circ }=\frac{1}{4}[/latex]. 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 [latex]\frac{360^\circ }{360^\circ }=1[/latex]. 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.

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°.

**Figure 7.** Quadrantal angles have a terminal side that lies along an axis. Examples are shown.

### A GENERAL NOTE: QUADRANTAL ANGLES

Quadrantal angles are angles whose 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 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.

Show Solution

### TRY IT 1

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

Show Solution

Watch this video for more examples of determining angles of 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 [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 to the radius, which is always [latex]2\pi[/latex] 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.

**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. So

[latex]\begin{gathered} 2\pi \text{ radians}={360}^{\circ } \\ \pi \text{ radians}=\frac{{360}^{\circ }}{2}={180}^{\circ } \\ 1\text{ radian}=\frac{{180}^{\circ }}{\pi }\approx {57.3}^{\circ } \end{gathered}[/latex]

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 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** [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 as the ratio of the arc length [latex]s[/latex] to the radius [latex]r[/latex].

[latex]\begin{gathered}s=r\theta \\ \theta =\frac{s}{r}\end{gathered}[/latex]

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

**Figure 12.**(a) In an angle of 1 radian, the arc length [latex]s[/latex] equals the radius [latex]r[/latex]. (b) An angle of 2 radians has an arc length [latex]s=2r[/latex]. (c) A full revolution is [latex]2\pi [/latex] 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 [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 45° angle on the two circles, as inFigure 13.

**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.

[latex]\begin{gathered}\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{gathered}[/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.

### A GENERAL NOTE: 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 [latex]2\pi [/latex] radians. A half revolution (180°) 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.

**Q & A**

### A MEASURE OF 1 RADIAN LOOKS TO BE ABOUT 60°. IS THAT CORRECT?

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

## 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, [latex]C=2\pi r[/latex], and for the unit circle [latex]C=2\pi [/latex]. These two different ways to rotate around a circle give us a way to convert from degrees to radians.

[latex]\begin{gathered}1\text{ rotation }=360^\circ =2\pi \text{radians} \\ \frac{1}{2}\text{ rotation}=180^\circ =\pi \text{radians} \\ \frac{1}{4}\text{ rotation}=90^\circ =\frac{\pi }{2} \text{radians} \end{gathered}[/latex]

### Try It

## 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.

**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.

**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.

Show Solution

### TRY IT 3

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

Show Solution

## 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.

[latex]\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }[/latex]

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

[latex]\frac{\text{Degrees}}{180}=\frac{\text{Radians}}{\pi }[/latex]

## Converting between Radians and Degrees

To convert between degrees and radians, use the proportion

[latex]\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }[/latex]

### EXAMPLE 3: CONVERTING RADIANS TO DEGREES

Convert each radian measure to degrees.

a. [latex]\frac{\pi }{6}[/latex]

b. 3

Show Solution

### TRY IT 4

Convert [latex]-\frac{3\pi }{4}[/latex] radians to degrees.

Show Solution

### Try It

### EXAMPLE 4: CONVERTING DEGREES TO RADIANS

Convert [latex]15[/latex] degrees to radians.

Show Solution

### TRY IT 6

Convert 126° to radians.

Show Solution

### Try It

Watch the following video for an explanation of radian measure and examples of converting between radians and degrees.

## 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 [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 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.

**Figure 16.** An angle of 140° and an angle of –220° are coterminal angles.

This video shows examples of how to determine if two angles are coterminal.

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°. 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 [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 17for examples of reference angles for angles in different quadrants.

**Figure 17**

### A GENERAL NOTE: 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, [latex]{t}^{\prime }[/latex], formed by the terminal side of the angle [latex]t[/latex]and the horizontal axis.

### HOW TO: GIVEN AN ANGLE GREATER THAN 360°, FIND A COTERMINAL ANGLE BETWEEN 0° AND 360°.

- Subtract 360° from the given angle.
- If the result is still greater than 360°, subtract 360° again till the result is between 0° and 360°.
- 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 [latex]\theta [/latex] that is coterminal with an angle measuring 800°, where [latex]0^\circ \le \theta <360^\circ [/latex].

Show Solution

### TRY IT 8

Find an angle [latex]\alpha [/latex] that is coterminal with an angle measuring 870°, where [latex]0^\circ \le \alpha <360^\circ [/latex].

Show Solution

### HOW TO: GIVEN AN ANGLE WITH MEASURE LESS THAN 0°, FIND A COTERMINAL ANGLE HAVING A MEASURE BETWEEN 0° AND 360°.

- Add 360° to the given angle.
- If the result is still less than 0°, add 360° again until the result is between 0° and 360°.
- 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 [latex]\alpha [/latex] such that 0° ≤ *α* < 360°.

Show Solution

Watch this video for another example of how to determine positive and negative coterminal angles.

### TRY IT 9

Find an angle [latex]\beta [/latex] that is coterminal with an angle measuring −300° such that [latex]0^\circ \le \beta <360^\circ [/latex].

Show Solution

### Try It

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

- Subtract [latex]2\pi [/latex] from the given angle.
- 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].
- The resulting angle is coterminal with the original angle.

### EXAMPLE 7: 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].

Show Solution

### TRY IT 11

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].

Show Solution

### Try It

## 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.

### A General Note: 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 20, is

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

**Figure 20**

### How To: Given a circle of radius [latex]r[/latex], calculate the length [latex]s[/latex] of the arc subtended by a given angle of measure [latex]\theta [/latex].

- If necessary, convert [latex]\theta [/latex] to radians.
- Multiply the radius [latex]r[/latex] by the radian measure of [latex]\theta :s=r\theta [/latex].

### Example 8: 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.

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

Show Solution

### Try It

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

Show Solution

### Try It

Finding the Area of a Sector of a Circle

**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.)

[latex]\begin{align}\text{Area of sector}&=\left(\frac{\theta }{2\pi }\right)\pi {r}^{2} \\ &=\frac{\theta \pi {r}^{2}}{2\pi } \\ &=\frac{1}{2}\theta {r}^{2} \end{align}[/latex]

### A General Note: 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]

**Figure 21.** The area of the sector equals half the square of the radius times the central angle measured in radians.

### How To: Given a circle of radius [latex]r[/latex], find the area of a sector defined by a given angle [latex]\theta [/latex].

- If necessary, convert [latex]\theta [/latex] to radians.
- Multiply half the radian measure of [latex]\theta [/latex] by the square of the radius [latex]r:\text{ } A=\frac{1}{2}\theta {r}^{2}[/latex].

### Example 9: Finding the Area of a Sector

An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 22. What is the area of the sector of grass the sprinkler waters?

**Figure 22.** The sprinkler sprays 20 ft within an arc of 30°.

Show Solution

### Try It

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.

Show Solution

In the following video you will see how to calculate arc length and area of a sector of a circle.

## 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.

[latex]v=\frac{s}{t}[/latex]

**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. 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.

[latex]\omega =\frac{\theta }{t}[/latex]

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:

[latex]\begin{align}s&=r\theta \\ &=r\omega t \end{align}[/latex]

Substituting this into the linear speed equation gives:

[latex]\begin{align} v&=\frac{s}{t} \\ &=\frac{r\omega t}{t} \\ &=r\omega \end{align}[/latex]

### A General Note: Angular and Linear Speed

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

[latex]\omega =\frac{\theta }{t}[/latex]

The **linear speed**. [latex]v[/latex], of the point can be found as the distance traveled, arc length [latex]s[/latex], per unit time, [latex]t[/latex].

[latex]v=\frac{s}{t}[/latex]

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

[latex]v=r\omega[/latex]

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

### How To: Given the amount of angle rotation and the time elapsed, calculate the angular speed.

- If necessary, convert the angle measure to radians.
- Divide the angle in radians by the number of time units elapsed: [latex]\omega =\frac{\theta }{t}[/latex].
- The resulting speed will be in radians per time unit.

### Example 10: Finding Angular Speed

A water wheel, shown in Figure 23, completes 1 rotation every 5 seconds. Find the angular speed in radians per second.

**Figure 23**

Show Solution

### Try It

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.

Show Solution

### Try It

### How To: Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed.

- Convert the total rotation to radians if necessary.
- Divide the total rotation in radians by the elapsed time to find the angular speed: apply [latex]\omega =\frac{\theta }{t}[/latex].
- 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 [latex]v=r\mathrm{\omega}[/latex].

### Example 11: 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.

Show Solution

### Try It

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.

Show Solution

## Key Equations

arc length | [latex]s=r\theta [/latex] |

area of a sector | [latex]A=\frac{1}{2}\theta {r}^{2}[/latex] |

angular speed | [latex]\omega =\frac{\theta }{t}[/latex] |

linear speed | [latex]v=\frac{s}{t}[/latex] |

linear speed related to angular speed | [latex]v=r\omega [/latex] |

## 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 [latex]\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }[/latex].
- Two angles that have the same terminal side are called coterminal angles.
- We can find coterminal angles by adding or subtracting 360° or [latex]2\pi [/latex].
- 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.

## 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 [latex]\frac{\theta }{2\pi }[/latex] 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

- quadrantal angle
- an angle whose terminal side lies on an axis

- radian measure
- the ratio of the arc length formed by an angle divided by the radius of the circle

- radian
- 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

## FAQs

### Angles | Precalculus? ›

Types of Angles - **Acute, Obtuse, Right, Straight, Reflex**.

**What are the 5 types of angles? ›**

Types of Angles - **Acute, Obtuse, Right, Straight, Reflex**.

**What are the basics of angles? ›**

The names of basic angles are **Acute angle, Obtuse angle, Right angle, Straight angle, reflex angle and full rotation**. An angle is geometrical shape formed by joining two rays at their end-points. An angle is usually measured in degrees. There are various types of angles in geometry.

**What is a 78 degree angle called? ›**

Angles between 0 and 90 degrees (0°< θ <90°) are called acute angles. • Angles between 90 and 180 degrees (90°< θ <180°) are known as **obtuse angles**.

**What are the 4 main angles? ›**

Angles: **Acute, Obtuse, Straight and Right**

There are four types of angles depending on their size in degrees.

**What are the 4 main types of angles? ›**

**The different types of angles based on their measurements are:**

- Acute Angle - An angle less than 90 degrees.
- Right Angle - An angle that is exactly 90 degrees.
- Obtuse Angle - An angle more than 90 degrees and less than 180 degrees.
- Straight Angle - An angle that is exactly 180 degrees.

**What are the 3 rules for angles? ›**

**Angle Facts – GCSE Maths – Geometry Guide**

- Angles in a triangle add up to 180 degrees. ...
- Angles in a quadrilateral add up to 360 degrees. ...
- Angles on a straight line add up to 180 degrees. ...
- Opposite Angles Are Equal. ...
- Exterior angle of a triangle is equal to the sum of the opposite interior angles. ...
- Corresponding Angles are Equal.

**What is simplest angle? ›**

The smallest angle is 1. It is smaller than a right angle, so we call this an **acute angle**. The largest angle is 2. It is larger than a right angle, so we call this an obtuse angle.

**What is a 100 degree angle called? ›**

An **obtuse angle** is a type of angle whose degree measurement is more than 90° but less than 180°. Examples of obtuse angles are: 100°, 120°, 140°, 160°, 170°, etc.

**What is a 250 degree angle called? ›**

A reflex angle is an angle that is more than 180 degrees and less than 360 degrees.

### What is a 110 degree angle called? ›

An angle that measures greater than 90∘ and less than 180∘ is called an **obtuse angle**. Since 110∘ is an obtuse angle, the triangle is obtuse.

**What are 7 angles called? ›**

**A heptagon** is a polygon with 7 sides and 7 angles. Sometimes the heptagon is also known as “septagon”.

**What is a 60 degree angle called? ›**

What do You Call a 60-Degree Angle? An angle whose measure is more than 0° but less than 90° is called an acute angle. Angles measuring 30°, 40°, 60° are all acute angles. Therefore a 60-degree angle is known as an acute angle.

**What kind of angle is 10? ›**

**Acute angles** measure less than 90 degrees. Right angles measure 90 degrees. Obtuse angles measure more than 90 degrees.

**What are the 5 acute angles? ›**

Acute Angle Degree

The degree of an acute angle measures less than 90 degrees, i.e. less than a right angle. The examples of acute angle degrees are **12°, 35°, 48°, 65°, 80°, 89°**.

**What are the 7 types of angles with examples? ›**

**These various types of angles are given below;**

- Acute Angle.
- Obtuse Angle.
- Right Angle.
- Straight Angle.
- Reflex Angle.
- Full Angle.
- Zero Angle.

**What is the 3 4 5 angle rule? ›**

The 3:4:5 triangle is the best way I know to determine with absolutely certainty that an angle is 90 degrees. This rule says that **if one side of a triangle measures 3 and the adjacent side measures 4, then the diagonal between those two points must measure 5 in order for it to be a right triangle**.

**What are F angles called? ›**

**Corresponding angles**

These are sometimes known as 'F' angles. The diagram below shows parallel lines being intersected by another line. The two angles marked in this diagram are called corresponding angles and are equal to each other.

**What is the common angle rule? ›**

**If two angles adjacent to a common angle are congruent, then the overlapping angles formed are congruent**.

**How do you teach angles in a fun way? ›**

**Masking Tape on Tables**

Tape random straight lines across students' tables to create lots of angles where the tape overlaps. Then ask your students to sit around the table with a marker, and encourage them to classify as many angles as they could. After classifying angles, your pupils can then move on to measuring them.

### What is an angle for Grade 7? ›

An angle is **the space between two lines that intersect each other**. Terms and angles you should know: An acute angle is an angle less then 90°. A right angle is a 90°angle. An obtuse angle is more than 90°but less then 180° A straight angle is a 180°angle.

**What do you call a flat angle? ›**

In geometry, a **straight angle** is an angle, whose vertex point has a value of 180 degrees. Basically, it forms a straight line, whose sides lie in opposite directions from the vertex. It is also termed as “flat angles”

**Where is the smallest angle? ›**

The angle opposite the smallest side of a triangle has the smallest measure.

**What is zero angle? ›**

Explain the zero angle.

Zero angles are **angles with a measurement of zero degrees or zero radian**. The zero angle is an angle with a measure of regardless of the measuring unit. A angle is generated when two angle arms rest on top of each other with a common vertex.

**What is angel in maths? ›**

What is an angle? In Plane Geometry, **a figure which is formed by two rays or lines that shares a common endpoint** is called an angle. The word “angle” is derived from the Latin word “angulus”, which means “corner”. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

**What is a 360 degree angle called? ›**

An angle of measure 360 degree is called full angle.

An angle which measures more than 180∘ but less than 360∘ is called an obtuse angle.

**What is a 400 degree angle called? ›**

A unit of angular measure in which the angle of an entire circle is 400 gradians. A right angle is therefore 100 gradians. A gradian is sometimes also called a gon or a grade.

**What is a 182 degree angle called? ›**

Type of Angle | Description |
---|---|

Acute Angle | is less than 90° |

Right Angle | is 90° exactly |

Obtuse Angle | is greater than 90° but less than 180° |

Straight Angle | is 180° exactly |

**What do you call a 170 degree angle? ›**

angles greater than 90 degrees and less than 180 degrees are known as **obtuse angles**.. hence 170 degrees is an example of obtuse angle..

**What are 120 angles called? ›**

Angle greater than 90∘ is known as an **obtuse angle**. Angle smaller than 90∘ is known as an acute angle. Angle equal to 90∘ is known as a right angle. Since ∠ABC = 120∘ , it is an obtuse angle.

### What angle is 1800 degrees? ›

**The sum of all the interior angles of the dodecagon** is equal to 1800°.

**What is a 135 degree angle called? ›**

An angle measuring 135° is an **obtuse angle**.

**What are the 4 ways to name an angle? ›**

Certain lower-case letters of the English alphabet like a, b, and c, and Greek letters like alpha (α), beta (β), gamma (γ), and theta (θ) are also used to represent angles.

**What angle is 35 degrees? ›**

Hence, is an **acute angle**.

**What type of angle is 41 degrees? ›**

TYPES OF ANGLES | |
---|---|

ACUTE ANGLE | More than 0° Less than 90° |

RIGHT ANGLE | Equals to 90° |

OBTUSE ANGLE | More than 90° Less than 180° |

STRAIGHT ANGLE | Equals to 180° |

**Which angle equals 91? ›**

An angle measuring 91o is **obtuse angled**, as the angle measure is more than 90o.

**What type of angle is 164? ›**

The Obtuse Angles ClipArt gallery offers 89 illustrations of angles ranging from 91 degrees to 179 degrees. Obtuse angles are those that measure greater than a right angle (90 degrees) and less than a straight angle (180 degrees).

**What type of angle is 89 degrees? ›**

**Acute angle** is an angle that measures less than 90∘. So, an angle measuring 89∘ is acute angle.

**What are angles around a point? ›**

What are angles around a point? Angles around a point describes **the sum of angles that can be arranged together so that they form a full turn**. Angles around a point add to 360 °.

**What are the 7 types of angles? ›**

**Types of angles based on pairs**

- Adjacent Angles. If two angles have a common side and a common vertex, they are adjacent angles. ...
- Complementary Angles. ...
- Supplementary Angles. ...
- Vertical Angles. ...
- Alternate Interior Angles. ...
- Alternate Exterior Angles. ...
- Corresponding Angles.

### What are 5 different acute angles? ›

The degree of an acute angle measures less than 90 degrees, i.e. less than a right angle. The examples of acute angle degrees are **12°, 35°, 48°, 65°, 80°, 89°**.

**What are 8 angles? ›**

In geometry, **Octagon is a polygon that has 8 sides and 8 angles**. That means the number of vertices and edges of an octagon is 8, respectively. In simple words, the octagon is an 8-sided polygon, also called 8-gon, in a two-dimensional plane.

**What are the different types of angles and their functions? ›**

The angles are classified under the following types: **Acute Angle – an angle measure less than 90 degrees**. **Right Angle – an angle is exactly at 90 degrees**. **Obtuse Angle – an angle whose measure is greater than 90 degrees and less than 180 degrees**.

**What is an obtuse angle? ›**

Obtuse angle is **any angle greater than 90°**: Straight angle is an angle measured equal to 180°: Zero angle is an angle measured equal to 0°: Complementary angles are angles whose measures have a sum equal to 90°: Supplementary angles are angles whose measures have a sum equal to 180°.

**What does obtuse angle look like? ›**

What does an Obtuse Angle Look Like? The angles measuring greater than 90° and less than 180° are called obtuse angles in geometry. The obtuse angle lies between 90° and 180° and looks like **a reclined chair, an angle below the staircase, or an angle formed between a minute and an hour hand of a clock at 10:15 a.m.**

**What kind of angle is 10 degrees? ›**

**Acute angles** measure less than 90 degrees. Right angles measure 90 degrees. Obtuse angles measure more than 90 degrees.

**What are Z shaped angles called? ›**

Corresponding angles

The diagram below shows parallel lines being intersected by another line. The two angles marked in this diagram are called corresponding angles and are equal to each other. The two angles marked in each diagram below are called **alternate angles** or Z angles.

**What is the golden rule angles? ›**

We call the function Φ(α) = Φ1(𝛼) with this graph the general golden ratio function, or the GGR function. For this function, the minimum is 0.6180 at the angle-point 𝜋, and the maximum is 1.6180 at 0 and 2𝜋. **The Golden ratio equals to 1 at angles 2/3𝜋 and 4/3𝜋** (as shown in Fig.

**What are angles 3 and 5 called? ›**

**Corresponding Angles** Theorem. 5. ∠3 and ∠5 are supplementary ∠4 and ∠6 are supplementary.

**What are the 3 undefined terms of geometry? ›**

In geometry, **point, line, and plane** are considered undefined terms because they are only explained using examples and descriptions. that lie on the same line.

### What are angles 1 and 5 called? ›

∠1 and ∠5 are **corresponding angles**, so they have equal measures.