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Degrees and radians
Two ways to measure an angle, and converting between them.
A full turn is 360° or 2π radians, so 180° = π. Radians measure an angle by the arc it cuts on a unit circle, which is why calculus prefers them: with radians, the derivative of sin is exactly cos.
Drawing 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 can be named as ray EF, or in symbol form \(\overset{\to}{EF}.\)
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 is formed from \(\overset{\to}{ED}\) and \(\overset{\to}{EF}.\) Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form \(\text{∠}DEF.\)
Greek letters are often used as variables for the measure of an angle. is a list of Greek letters commonly used to represent angles, and a sample angle is shown in .
| \(\theta\) | \(\phi \ \text{or}\ ϕ\) | \(\alpha\) | \(\beta\) | \(\gamma\) |
| 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 .
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 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 .
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.
Condensed — the full section is in OpenStax Precalculus 2e.
Converting Between Degrees and Radians
Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.
The circumference of a circle is \(C=2\pi r.\) If we divide both sides of this equation by \(r,\) we create the ratio of the circumference to the radius, which is always \(2\pi\) regardless of the length of the radius. So the circumference of any circle is \(2\pi \approx 6.28\) times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in .
This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals \(2\pi\) times the radius, a full circular rotation is \(2\pi\) radians. So
\[\begin{array}{l}\begin{array}{l} \\ 2\pi \text{ radians}={360}^{∘}\end{array} \\ \pi \text{ radians}=\frac{{360}^{∘}}{2}={180}^{∘} \\ 1\text{ radian}=\frac{{180}^{∘}}{\pi }\approx {57.3}^{∘}\end{array}\]See . 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.
Condensed — the full section is in OpenStax Precalculus 2e.
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 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°. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.
An angle’s reference angle is the measure of the smallest, positive, acute angle \(t'\) formed by the terminal side of the angle \(t\) and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See for examples of reference angles for angles in different quadrants.
Condensed — the full section is in OpenStax Precalculus 2e.
Determining the Length of an Arc
Recall that the radian measure \(\theta\) of an angle was defined as the ratio of the arc length \(s\) of a circular arc to the radius \(r\) of the circle, \(\theta =\frac{s}{r}.\) From this relationship, we can find arc length along a circle, given an angle.
Example
Try it.
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.
Solution
- ⓐLet’s begin by finding the circumference of Mercury’s orbit.
\[\begin{array}{l}C=2\pi r \\ =2\pi (36\text{ million miles}) \\ \approx 226\text{ million miles}\end{array}\]
Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:
\[(0.0114)226\text{ million miles = 2}\text{.58 million miles}\] - ⓑ Now, we convert to radians: \[\begin{array}{l}\text{radian} \\ =\frac{\text{arclength}}{\text{radius}} \\ =\frac{2.\text{58 million miles}}{36\text{ million miles}} \\ =0.0717\end{array}\]
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 \(r\) can be found using the formula \(A=\pi {r}^{2}.\) If the two radii form an angle of \(\theta ,\) measured in radians, then \(\frac{\theta }{2\pi }\) is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction \(\frac{\theta }{2\pi }\) multiplied by the entire area. (Always remember that this formula only applies if \(\theta\) is in radians.)
\[\begin{array}{l}\text{Area of sector} \\ =(\frac{\theta }{2\pi })\pi {r}^{2} \\ =\frac{\theta \pi {r}^{2}}{2\pi } \\ =\frac{1}{2}\theta {r}^{2}\end{array}\]Example
Try it.
An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in . What is the area of the sector of grass the sprinkler waters?
Solution
First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:
\[\begin{array}{l}30\text{ degrees}=30⋅\frac{\pi }{180} \\ =\frac{\pi }{6}\text{ radians}\end{array}\]The area of the sector is then
\[\begin{array}{l}\text{Area = }\frac{1}{2}(\frac{\pi }{6}){(20)}^{2} \\ \approx 104.72\end{array}\]So the area is about \(104.72{\text{ ft}}^{2}.\)
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 \(10\pi\) inches, every second. So the linear speed of the point is \(10\pi\) in./s. The equation for linear speed is as follows where \(v\) is linear speed, \(s\) is displacement, and \(t\) is time.
\[v=\frac{s}{t}\]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, \(\ \theta \\) is the angle traversed, and \(\ t\\) is time.
\[\omega =\frac{\theta }{t}\]Combining the definition of angular speed with the arc length equation, \(s=r\theta ,\) we can find a relationship between angular and linear speeds. The angular speed equation can be solved for \(\theta ,\) giving \(\theta =\omega t.\) Substituting this into the arc length equation gives:
\[\begin{array}{l}s=r\theta \\ =r\omega t\end{array}\]Substituting this into the linear speed equation gives:
\[\begin{array}{l}\begin{array}{l} \\ v=\frac{s}{t}\end{array} \\ =\frac{r\omega t}{t} \\ =r\omega \end{array}\]Condensed — the full section is in OpenStax Precalculus 2e.
Key Equations
| arc length | \(s=r\theta\) |
| area of a sector | \(A=\frac{1}{2}\theta {r}^{2}\) |
| angular speed | \(\omega =\frac{\theta }{t}\) |
| linear speed | \(v=\frac{s}{t}\) |
| linear speed related to angular speed | \(v=r\omega\) |
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. See .
- In addition to degrees, the measure of an angle can be described in radians. See .
- To convert between degrees and radians, use the proportion \(\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }.\) See and .
- Two angles that have the same terminal side are called coterminal angles.
- We can find coterminal angles by adding or subtracting 360° or \(2\pi .\) See and .
- Coterminal angles can be found using radians just as they are for degrees. See .
- The length of a circular arc is a fraction of the circumference of the entire circle. See .
- The area of sector is a fraction of the area of the entire circle. See .
- 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. See .
- The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. See .
Eksempel på arbeid: 30 degrees to radians
Steg for trinn
- 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6} \approx 0.52360
Multiply by π/180 (a full turn is 360° = 2π).
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Practice (40)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
-
- ⓐ Sketch an angle of 30° in standard position.
- ⓑ Sketch an angle of −135° in standard position.
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- ⓐ Divide the angle measure by 360°.
\[\frac{30^{\circ}}{360^{\circ}}=\frac{1}{12}\]
To rewrite the fraction in a more familiar fraction, we can recognize that
\[\frac{1}{12}=\frac{1}{3}(\frac{1}{4})\]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 .
- ⓑ Divide the angle measure by 360°.
\[\frac{-135^{\circ}}{360^{\circ}}=-\frac{3}{8}\]
In this case, we can recognize that
\[-\frac{3}{8}=-\frac{3}{2}(\frac{1}{4})\]Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in .
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Show an angle of 240° on a circle in standard position.
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Find the radian measure of one-third of a full rotation.
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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,
\[\begin{array}{l} \\ \begin{array}{l}s=\frac{1}{3}(2\pi r) \\ =\frac{2\pi r}{3}\end{array}\end{array}\]The radian measure would be the arc length divided by the radius.
\[\begin{array}{l}\begin{array}{l}\begin{array}{l}\text{radian measure} \\ =\frac{\frac{2\pi r}{3}}{r} \\ =\frac{2\pi r}{3r}\end{array} \\ =\frac{2\pi }{3}\end{array} \\ \\ \end{array}\] -
Find the radian measure of three-fourths of a full rotation.
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\(\frac{3\pi }{2}\)
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Convert each radian measure to degrees.
- ⓐ \(\frac{\pi }{6}\)
- ⓑ 3
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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. \[\begin{array}{l}\begin{array}{l} \\ \frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }\end{array} \\ \frac{\theta }{180}=\frac{\frac{\pi }{6}}{\pi } \\ \theta =\frac{180}{6} \\ \theta ={30}^{∘}\end{array}\]
- ⓑ We use the proportion, substituting the given information. \[\begin{array}{l}\begin{array}{l} \\ \frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }\end{array} \\ \frac{\theta }{180}=\frac{3}{\pi } \\ \theta =\frac{3(180)}{\pi } \\ \theta \approx {172}^{∘}\end{array}\]
-
Convert \(-\frac{3\pi }{4}\) radians to degrees.
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−135°
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Convert \(15\) degrees to radians.
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In this example, we start with degrees and want radians, so we again set up a proportion and solve it, but we substitute the given information into a different part of the proportion.
\[\begin{array}{l}\begin{array}{l} \\ \frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }\end{array} \\ \frac{15}{180}=\frac{{\theta }^{R}}{\pi } \\ \frac{15\pi }{180}={\theta }^{R} \\ \frac{\pi }{12}={\theta }^{R}\end{array}\] -
Convert 126° to radians.
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\(\frac{7\pi }{10}\)
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Find the least positive angle \(\theta\) that is coterminal with an angle measuring 800°, where \(0^{\circ}\le \theta <360^{\circ}.\)
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An angle with measure 800° is coterminal with an angle with measure 800 − 360 = 440°, but 440° is still greater than 360°, so we subtract 360° again to find another coterminal angle: 440 − 360 = 80°.
The angle \(\theta =80^{\circ}\) is coterminal with 800°. To put it another way, 800° equals 80° plus two full rotations, as shown in .
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Find an angle \(\alpha\) that is coterminal with an angle measuring 870°, where \(0^{\circ}\le \alpha <360^{\circ}.\)
Vis svaret
\(\alpha =150^{\circ}\)
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Show the angle with measure −45° on a circle and find a positive coterminal angle \(\alpha\) such that 0° ≤ α < 360°.
Vis svaret
Since 45° is half of 90°, we can start at the positive horizontal axis and measure clockwise half of a 90° angle.
Because we can find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°:
\[-45^{\circ}+360^{\circ}=315^{\circ}\]We can then show the angle on a circle, as in .
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Find an angle \(\beta\) that is coterminal with an angle measuring −300° such that \(0^{\circ}\le \beta <360^{\circ}.\)
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\(\beta =60^{\circ}\)
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Find an angle \(\beta\) that is coterminal with \(\frac{19\pi }{4},\) where \(0\le \beta <2\pi .\)
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When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of \(\ 2\pi \\) radians:
\[\begin{array}{l}\frac{19\pi }{4}-2\pi =\frac{19\pi }{4}-\frac{8\pi }{4} \\ =\frac{11\pi }{4}\end{array}\]The angle \(\frac{11\pi }{4}\) is coterminal, but not less than \(2\pi ,\) so we subtract another rotation:
\[\begin{array}{l}\frac{11\pi }{4}-2\pi =\frac{11\pi }{4}-\frac{8\pi }{4} \\ =\frac{3\pi }{4}\end{array}\]The angle \(\frac{3\pi }{4}\) is coterminal with \(\frac{19\pi }{4},\) as shown in .
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Find an angle of measure \(\theta\) that is coterminal with an angle of measure \(-\frac{17\pi }{6}\) where \(0\le \theta <2\pi .\)
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\(\frac{7\pi }{6}\)
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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.
Vis svaret
- ⓐLet’s begin by finding the circumference of Mercury’s orbit.
\[\begin{array}{l}C=2\pi r \\ =2\pi (36\text{ million miles}) \\ \approx 226\text{ million miles}\end{array}\]
Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:
\[(0.0114)226\text{ million miles = 2}\text{.58 million miles}\] - ⓑ Now, we convert to radians: \[\begin{array}{l}\text{radian} \\ =\frac{\text{arclength}}{\text{radius}} \\ =\frac{2.\text{58 million miles}}{36\text{ million miles}} \\ =0.0717\end{array}\]
-
Find the arc length along a circle of radius 10 units subtended by an angle of 215°.
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\(\frac{215\pi }{18}=37.525\text{ units}\)
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An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in . What is the area of the sector of grass the sprinkler waters?
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First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:
\[\begin{array}{l}30\text{ degrees}=30⋅\frac{\pi }{180} \\ =\frac{\pi }{6}\text{ radians}\end{array}\]The area of the sector is then
\[\begin{array}{l}\text{Area = }\frac{1}{2}(\frac{\pi }{6}){(20)}^{2} \\ \approx 104.72\end{array}\]So the area is about \(104.72{\text{ ft}}^{2}.\)
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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.
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1.88
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A water wheel, shown in , completes 1 rotation every 5 seconds. Find the angular speed in radians per second.
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The wheel completes 1 rotation, or passes through an angle of \(2\pi\) radians in 5 seconds, so the angular speed would be \(\omega =\frac{2\pi }{5}\approx 1.257\\) radians per second.
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A vintage 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.
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\(\frac{3\pi }{2}\) rad/s
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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.
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Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.
We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:
\[180\frac{\text{rotations}}{\text{minute}}⋅\frac{2\pi \ \text{radians}}{\text{rotation}}=360\pi \frac{\text{radians}}{\text{minute}}\]Using the formula from above along with the radius of the wheels, we can find the linear speed:
\[\begin{array}{l}\begin{array}{l} \\ v=(14\ \text{inches})(360\pi \frac{\text{radians}}{\text{minute}})\end{array} \\ =5040\pi \frac{\text{inches}}{\text{minute}}\end{array}\]Remember that radians are a unitless measure, so it is not necessary to include them.
Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.
\[5040\pi \frac{\text{inches}}{\text{minute}}⋅\frac{\text{1 }\text{feet}}{\text{12 }\text{inches}}⋅\frac{\text{1 mile}}{\text{5280 }\text{feet}}⋅\frac{\text{60 }\text{minutes}}{\text{1 hour}}\approx 14.99\ \text{miles per hour (mph)}\] -
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.
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1655 kilometers per hour
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Draw an angle in standard position. Label the vertex, initial side, and terminal side.
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Explain why there are an infinite number of angles that are coterminal to a certain angle.
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State what a positive or negative angle signifies, and explain how to draw each.
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Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.
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How does radian measure of an angle compare to the degree measure? Include an explanation of 1 radian in your paragraph.
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Explain the differences between linear speed and angular speed when describing motion along a circular path.
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Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.
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\(\frac{2\pi }{3}\)
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\(\frac{7\pi }{4}\)
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\(\frac{5\pi }{6}\)
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\(\frac{\pi }{2}\)
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\(-\frac{\pi }{10}\)
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\(\frac{22\pi }{3}\)
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\(\frac{4\pi }{3}\)
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\(-\frac{\pi }{6}\)
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\(-\frac{4\pi }{3}\)
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\(\frac{2\pi }{3}\)
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Find the arc length.
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Find the area of the sector.
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\(\frac{7\pi }{2}\approx 11.00{\text{ in}}^{2}\)
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Find the arc length.
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Find the area of the sector.
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\(\frac{81\pi }{20}\approx 12.72{\text{ cm}}^{2}\)
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\(\frac{3\pi }{4}\) radians
Symbols used here
Ratio of a circle's circumference to its diameter, 3.14159…
The usual name for an angle.
1/360 of a full turn. 180° = π radians.
Equal to the precision shown, not exactly.
Inequalities that allow equality; < and > exclude it.
Ratios of sides in a right triangle; coordinates on the unit circle.
The angle whose sine is the given value (and likewise arccos, arctan).
How to: Degrees and radians
- 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.
- 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
Questions people ask
Why radians instead of degrees?
A radian is the angle whose arc equals the radius, so it is a pure ratio rather than an arbitrary 1/360 of a turn. With radians the derivative of sin x is exactly cos x; with degrees a factor of π/180 appears everywhere.
Why does sin x = 1/2 have infinitely many solutions?
Sine repeats every full turn, and within one turn it reaches 1/2 twice (at π/6 and 5π/6). Add any whole number of turns to either and it is still true.
How do I remember the exact values?
Two triangles: the 45-45-90 with sides 1, 1, √2 and the 30-60-90 with sides 1, √3, 2. Every value for 30°, 45° and 60° is a ratio of those sides; symmetry gives the rest of the circle.
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Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.
Mer i Trigonometry
The unit circleTrigonometric equationsTrigonometric identitiesRight-triangle trigonometry (SOH-CAH-TOA)Law of sines and law of cosinesGraphs of sine, cosine and tangentInverse trigonometric functions