| title | layout |
|---|---|
Angles |
page |
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.
Properly defining an angle first requires that we define a ray. A ray{: data-type="term"} consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint{: data-type="term" .no-emphasis} of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in [link] can be named as ray EF, or in symbol form EF→.
{: #Figure_05_01_001}
An angle{: data-type="term"} is the union of two rays having a common endpoint. The endpoint is called the vertex{: data-type="term"} of the angle, and the two rays are the sides of the angle. The angle in [link] is formed from ED →
and EF → .
Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form ∠DEF.
{: #Figure_05_01_002}
Greek letters are often used as variables for the measure of an angle. [link] is a list of Greek letters commonly used to represent angles, and a sample angle is shown in [link].
| θ
| φorϕ
| α
| β
| γ
| | theta | phi | alpha | beta | gamma | {: #Table_05_01_01 summary="Two rows and five columns. First row shows symbols for theta, phi, alpha, beta, and gamma. Second row spells out name for each symbol."}
{: #Figure_05_01_003}
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{: data-type="term"}, and the rotated ray is the terminal side{: data-type="term"}. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in [link].
{: #Figure_05_01_004}
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{: data-type="term"} 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{: data-type="term"} is 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{: data-type="term"} if its vertex is located at the origin, and its initial side extends along the positive x-axis. See [link].
{: #Figure_05_01_005}
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{: data-type="term"}. If the angle is measured in a clockwise direction, the angle is said to be a negative angle{: data-type="term"}.
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 90° 360° = 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 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 [link].
{: #Figure_05_01_006}
Since we define an angle in standard position{: data-type="term" .no-emphasis} by its initial 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 [link].
{: #Figure_05_01_018}
- 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. {: type="1"}
To rewrite the fraction in a more familiar fraction, we can recognize that
<div data-type="equation" id="eip-id1165135613636" class="unnumbered" data-label="">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mfrac> <mn>1</mn> <mrow> <mn>12</mn></mrow> </mfrac> <mo>=</mo><mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow><mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo></mrow></mrow> </math>
</div>
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 [[link]](#Figure_05_01_007).
{: #Figure_05_01_007}
-
Divide the angle measure by 360°.
−135° 360° =− 3 8In this case, we can recognize that
− 3 8 =− 3 2 ( 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 [link].
{: #Figure_05_01_008}
{: type="a"}
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{: data-type="term" .no-emphasis} 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{: data-type="term" .no-emphasis} of that circle.
The circumference of a circle is C=2πr.
If we divide both sides of this equation by r,
we create the ratio of the circumference to the radius, which is always 2π
regardless of the length of the radius. So the circumference of any circle is 2π≈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 [link].
{: #Figure_05_01_010}
This brings us to our new angle measure. One radian{: data-type="term" .no-emphasis} 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 2π
times the radius, a full circular rotation is 2π
radians. So
See [link]. 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_05_01_011}
An arc length{: data-type="term"} 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{: data-type="term" .no-emphasis}, 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 [link].
If s=r,
then θ= r r = 1 radian.
{: #Figure_05_01_013}
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=2πr,
where r
is the radius. The smaller circle then has circumference 2π(2)=4π
and the larger has circumference 2π(3)=6π.
Now we draw a 45° angle on the two circles, as in [link].
{: #Figure_05_01_012}
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.
Since both ratios are 1 4 π,
the angle measures of both circles are the same, even though the arc length and radius differ.
radians. A half revolution (180°) is equivalent to π
radians.
The radian measure{: data-type="term"} 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 s r
radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.
Yes. It is approximately 57.3°. Because 2π radians equals 360°, 1 radian equals 360° 2π ≈57.3°.
Because radian{: data-type="term" .no-emphasis} measure is the ratio of two lengths, it is a unitless measure. For example, in [link], 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{: data-type="term" .no-emphasis} (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=2πr,
and for the unit circle C=2π.
These two different ways to rotate around a circle give us a way to convert from degrees to 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 [link]. Memorizing these angles will be very useful as we study the properties associated with angles.
{: #Figure_05_01_016}
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in [link], which are shown in [link]. Be sure you can verify each of these measures.
{: #Figure_05_01_017}
Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.
This proportion shows that the measure of angle θ
in degrees divided by 180 equals the measure of angle θ
in radians divided by π.
Or, phrased another way, degrees is to 180 as radians is to π.
-
π 6
-
3 {: type="a"}
-
We use the proportion, substituting the given information.
θ 180 = θ R π θ 180 = π 6 π θ= 180 6 θ= 30 ∘ -
We use the proportion, substituting the given information.
θ 180 = θ R π θ 180 = 3 π θ= 3(180) π θ≈ 172 ∘
{: type="a"}
radians to degrees.
degrees to radians.
Because 15 ∘ = 1 2 ( 30 ∘ ),
we can find that 1 2 ( π 6 )
is π 12 .
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 2π.
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 [link]. The angle of 140° is a positive angle{: data-type="term" .no-emphasis}, measured counterclockwise. The angle of –220° is a negative angle{: data-type="term" .no-emphasis}, 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{: data-type="term" .no-emphasis}. 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_05_01_020}
Any angle has infinitely many coterminal angles{: data-type="term" .no-emphasis} 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 [link] for examples of reference angles for angles in different quadrants.
{: #Figure_05_01_019}
An angle’s reference angle{: data-type="term"} is the size of the smallest acute angle, t ′ ,
formed by the terminal side of the angle t
and the horizontal axis.
- 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. {: type="1"}
that is coterminal with an angle measuring 800°, where 0°≤θ<360°.
The angle θ=80°
is coterminal with 800°. To put it another way, 800° equals 80° plus two full rotations, as shown in [link].
{: #Figure_05_01_021}
that is coterminal with an angle measuring 870°, where 0°≤α<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. {: type="1"}
such that 0° ≤ α < 360°.
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°:
{: #Figure_05_01_022}
that is coterminal with an angle measuring −300° such that 0°≤β<360°.
We can find coterminal angles{: data-type="term" .no-emphasis} 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.
find a coterminal angle between 0 and 2π.
-
Subtract 2π
from the given angle.
-
If the result is still greater than 2π,
subtract 2π
again until the result is between 0
and 2π.
-
The resulting angle is coterminal with the original angle. {: type="1"}
that is coterminal with 19π 4 ,
where 0≤β<2π.
radians:
is coterminal, but not less than 2π,
so we subtract another rotation:
is coterminal with 19π 4 ,
as shown in [link].
{: #Figure_05_01_023}
that is coterminal with an angle of measure − 17π 6
where 0≤θ<2π.
Recall that the radian measure{: data-type="term" .no-emphasis} θ
of an angle was defined as the ratio of the arc length{: data-type="term" .no-emphasis} s
of a circular arc to the radius r
of the circle, θ= s r .
From this relationship, we can find arc length along a circle, given an angle.
subtended by an angle with measure θ
in radians, shown in [link], is
calculate the length s
of the arc subtended by a given angle of measure θ.
-
If necessary, convert θ
to radians.
-
Multiply the radius r
by the radian measure of θ:s=rθ. {: type="1"}
- 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. {: type="a"}
Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:
<div data-type="equation" id="eip-id1165134109674" class="unnumbered" data-label="">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mrow><mo>(</mo> <mrow> <mn>0.0114</mn></mrow> <mo>)</mo></mrow><mn>226</mn><mtext> million miles = 2</mtext><mtext>.58 million miles</mtext></mrow> </math>
</div>
- Now, we convert to radians:
radian= arclength radius = 2.58 million miles 36 million miles =0.0717
{: type="a"}
In addition to arc length, we can also use angles to find the area of a sector of a circle{: data-type="term" .no-emphasis}. 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=π r 2 .
If the two radii form an angle of θ,
measured in radians, then θ 2π
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{: data-type="term"} is the fraction θ 2π
multiplied by the entire area. (Always remember that this formula only applies if θ
is in radians.)
subtended by an angle θ,
measured in radians, is
{: #Figure_05_01_026}
-
If necessary, convert θ
to radians.
-
Multiply half the radian measure of θ
by the square of the radius r: A= 1 2 θ r 2 . {: type="1"}
{: #Figure_05_01_027}
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{: data-type="term"}) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10π
inches, every second. So the linear speed of the point is 10π
in./s. The equation for linear speed is as follows where v
is linear speed, s
is displacement, and t
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 360 degrees 4 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 ω
(read as omega) is angular speed, θ
is the angle traversed, and t
is time.
Combining the definition of angular speed with the arc length equation, s=rθ,
we can find a relationship between angular and linear speeds. The angular speed equation can be solved for θ,
giving θ=ωt.
Substituting this into the arc length equation gives:
Substituting this into the linear speed equation gives:
its angular speed{: data-type="term"}, ω,
is the angular rotation θ
per unit time, t.
of the point can be found as the distance traveled, arc length s,
per unit time, t.
representing the amount of rotation occurring in a unit of time, can be multiplied by the radius r
to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.
-
If necessary, convert the angle measure to radians.
-
Divide the angle in radians by the number of time units elapsed: ω= θ t
.
-
The resulting speed will be in radians per time unit. {: type="1"}
{: #Figure_05_01_028}
radians in 5 seconds, so the angular speed would be ω= 2π 5 ≈1.257
radians per second.
rad/s
-
Convert the total rotation to radians if necessary.
-
Divide the total rotation in radians by the elapsed time to find the angular speed: apply ω= θ t .
-
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 v=rω. {: type="1"}
We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:
Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.
| arc length | s=rθ
| | area of a sector | A= 1 2 θ r 2
| | angular speed | ω= θ t
| | linear speed | v= s t
| | linear speed related to angular speed | v=rω
| {: #eip-id1165134342478 summary=".."}
-
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 [link].
-
In addition to degrees, the measure of an angle can be described in radians. See [link].
-
To convert between degrees and radians, use the proportion θ 180 = θ R π .
-
Two angles that have the same terminal side are called coterminal angles.
-
We can find coterminal angles by adding or subtracting 360° or 2π.
-
Coterminal angles can be found using radians just as they are for degrees. See [link].
-
The length of a circular arc is a fraction of the circumference of the entire circle. See [link].
-
The area of sector is a fraction of the area of the entire circle. See [link].
-
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 [link].
-
The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. See [link].
For the following exercises, draw an angle in standard position with the given measure.
For the following exercises, refer to [link]. Round to two decimal places.
{: #Figure_05_01_218}
For the following exercises, refer to [link]. Round to two decimal places.
{: #Figure_05_01_219}
For the following exercises, convert angles in radians to degrees.
radians
radians
radians
radians
radians
radians
radians
For the following exercises, convert angles in degrees to radians.
radians
radians
radians
radians
For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places.
radians.
miles
centimeters
meters
For the following exercises, use the given information to find the area of the sector. Round to four decimal places.
radians.
radians.
For the following exercises, find the angle between 0° and 360° that is coterminal to the given angle.
For the following exercises, find the angle between 0 and 2π
in radians that is coterminal to the given angle.
the angular speed in RPM, and the angular speed in rad/s?
the angular speed in RPM, and the angular speed in deg/s?
. The radius of Earth is 3960 miles.
miles
The radius of Earth is 3960
miles.
minutes?
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 θ 2π
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