Angles

A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.

Drawing Angles in Standard Position

Properly defining an angle first requires that we define a ray. A ray 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 below can be named as ray EF, or in symbol form $\overrightarrow{EF}$.

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

Greek letters are often used as variables for the measure of an angle. The table below is a list of Greek letters commonly used to represent angles, alongside a sample angle $\angle\theta$.

Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex, as below.

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 $\tfrac{1}{360}$ 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 below.

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.

Drawing an angle in standard position always starts the same way—draw the initial side along the positive $x$-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that $\tfrac{90^\circ}{360^\circ}=\tfrac{1}{4}$. 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 $\tfrac{360^\circ}{360^\circ}=1$. 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 below.

Since we define an angle in standard position 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 below.

Example. Sketch an angle of 30° in standard position, and sketch an angle of $-135^\circ$ in standard position.

Solution.

1. Divide the angle measure by 360°.

   $$\tfrac{30^\circ}{360^\circ}=\tfrac{1}{12}$$

   To rewrite the fraction in a more familiar fraction, we can recognize that

   $$\tfrac{1}{12}=\tfrac{1}{3}\left(\tfrac{1}{4}\right)$$

   One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30° as below.

2. Divide the angle measure by 360°.

   $$\tfrac{-135^\circ}{360^\circ}=-\tfrac{3}{8}$$

   In this case, we can recognize that

   $$-\tfrac{3}{8}=-\tfrac{3}{2}\left(\tfrac{1}{4}\right)$$

   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 below.

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 $2\pi\approx6.28$ 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 below.

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

See below. 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.

Relating Arc Lengths to Radius

An arc length $s$ is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.

This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length $s$ to the radius $r$. See below.

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

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is $C=2\pi r$, where $r$ is the radius. The smaller circle then has circumference $2\pi(2)=4\pi$ and the larger has circumference $2\pi(3)=6\pi$. Now we draw a 45° angle on the two circles, as below.

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

Since both ratios are $\tfrac{1}{4}\pi$, the angle measures of both circles are the same, even though the arc length and radius differ.

Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.

Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, $C=2\pi r$, and for the unit circle $C=2\pi$. These two different ways to rotate around a circle give us a way to convert from degrees to radians.

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

Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed above, which are shown below. Be sure you can verify each of these measures.

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

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

So,

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

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.

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

Example. Convert each radian measure to degrees.

1. $\tfrac{\pi}{6}$
2. $3$

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

1. We use the proportion, substituting the given information.

   $$\begin{array}{lrcl} & \tfrac{\theta}{180} &=& \tfrac{\theta^R}{\pi} \\[4pt] & \tfrac{\theta}{180} &=& \tfrac{\tfrac{\pi}{6}}{\pi} \\[4pt] & \theta &=& \tfrac{180}{6} \\[4pt] & \theta &=& 30^\circ \end{array}$$

2. We use the proportion, substituting the given information.

   $$\begin{array}{lrcl} & \tfrac{\theta}{180} &=& \tfrac{\theta^R}{\pi} \\[4pt] & \tfrac{\theta}{180} &=& \tfrac{3}{\pi} \\[4pt] & \theta &=& \tfrac{3(180)}{\pi} \\[4pt] & \theta &\approx& 172^\circ \end{array}$$

Example. Convert 15 degrees to radians.

Solution. 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.

Analysis. Another way to think about this problem is by remembering that $30^\circ=\tfrac{\pi}{6}$. Because $15^\circ=\tfrac{1}{2}(30^\circ)$, we can find that $\tfrac{1}{2}\left(\tfrac{\pi}{6}\right)$ is $\tfrac{\pi}{12}$.

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 figure below. The angle of 140° is a positive angle, measured counterclockwise. The angle of $-220^\circ$ 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 $-260^\circ$. 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 below for examples of reference angles for angles in different quadrants.

Example. Find the least positive angle $\theta$ that is coterminal with an angle measuring 800°, where $0^\circ\le\theta<360^\circ$.

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

The angle $\theta=80^\circ$ is coterminal with 800°. To put it another way, 800° equals 80° plus two full rotations, as shown below.

Example. Show the angle with measure $-45^\circ$ on a circle and find a positive coterminal angle $\alpha$ such that $0^\circ\le\alpha<360^\circ$.

Solution. 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°:

We can then show the angle on a circle, as below.

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.

Example. Find an angle $\beta$ that is coterminal with $\tfrac{19\pi}{4}$, where $0\le\beta<2\pi$.

Solution. 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:

The angle $\tfrac{11\pi}{4}$ is coterminal, but not less than $2\pi$, so we subtract another rotation:

The angle $\tfrac{3\pi}{4}$ is coterminal with $\tfrac{19\pi}{4}$, as shown below.

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, $\theta=\tfrac{s}{r}$. From this relationship, we can find arc length along a circle, given an angle.

Example. Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

1. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
2. Use your answer from part (1) to determine the radian measure for Mercury’s movement in one Earth day.

Solution.

1. Let’s begin by finding the circumference of Mercury’s orbit.

   $$\begin{array}{lrcl} & C &=& 2\pi r \\[4pt] & &=& 2\pi(36\text{ million miles}) \\[4pt] & &\approx& 226\text{ million miles} \end{array}$$

   Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:

   $$(0.0114)226\text{ million miles}=2.58\text{ million miles}$$

2. Now, we convert to radians:

   $$\begin{array}{lrcl} \text{radian} & &=& \tfrac{\text{arc length}}{\text{radius}} \\[4pt] & &=& \tfrac{2.58\text{ million miles}}{36\text{ million miles}} \\[4pt] & &=& 0.0717 \end{array}$$

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 $\tfrac{\theta}{2\pi}$ 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 $\tfrac{\theta}{2\pi}$ multiplied by the entire area. (Always remember that this formula only applies if $\theta$ is in radians.)

Example. An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown below. 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:

The area of the sector is then

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.

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 $\tfrac{360\text{ degrees}}{4\text{ seconds}}=$ 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.

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:

Substituting this into the linear speed equation gives:

Water wheels have been used for thousands of years to transfer the power of flowing water to other devices. Water turned the wheel, which in turn rotated a crank connected to two saws used to cut blocks. These design elements were used in water wheel applications throughout the world, and even provided the underlying principle for the steam engine, invented about 1500 years later.

Example. A water wheel completes 1 rotation every 5 seconds. Find the angular speed in radians per second.

Solution. The wheel completes 1 rotation, or passes through an angle of $2\pi$ radians in 5 seconds, so the angular speed would be $\omega=\tfrac{2\pi}{5}\approx1.257$ radians per second.

Example. 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.

Solution. 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:

Using the formula from above along with the radius of the wheels, we can find the linear speed:

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.

Key equations

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 $\tfrac{\theta}{180}=\tfrac{\theta^R}{\pi}$.
• Two angles that have the same terminal side are called coterminal angles.
• We can find coterminal angles by adding or subtracting 360° or $2\pi$.
• 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 a 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.

Practice

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

This section is adapted from Precalculus 2e, Section 5.1: Angles by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated all twenty-three instructional figures the source draws before its Section Exercises as accessible spec-first SVGs, thirty-two apfigure panels in all — ray EF; angle DEF; the sample angle θ; the initial-side/terminal-side/vertex diagram (its small rotation-indicator arc is omitted, since the figure engine’s arc primitive requires a coordinate frame that a bare, axis-free angle diagram does not carry — a simplification, not a content loss, since the initial/terminal/vertex labels already convey the rotation); the standard-position schematic; the 90°/360° drawing pair; the four quadrantal-angle panels; the 30° and −135° worked-example angles; the radius-string circle illustrating why $2\pi\approx6.28$; the one-radian construction; the three-panel 1-radian/2-radian/full-revolution figure (drawn with a full circle, three dashed diameters connecting antipodal radian markers, and one solid radius, in place of the source’s hand-drawn spiral of radius arcs); the 45°-on-two-circles figure (its decorative dashed crosshair, which carries no additional angle information beyond the labeled 45° ray itself, is omitted); the sixteen-ray common-angle wheels in degrees and radians; the 140°/−220° and −45°/315° coterminal pairs; the four reference-angle panels; the 800°/80° and 19π/4/3π/4 coterminal pairs (each drawn as an Archimedean spiral $r=r_0+(r_1-r_0)\tfrac{\theta}{\theta_{\max}}$ sampled from that equation at three-degree steps, so the two extra full rotations are shown as the source draws them, plus a labeled outer arc for the coterminal angle); the generic arc-length and sector-area diagrams; and the sprinkler sector. The two Section Exercises figures (radius 3 in at 140°; radius 4.5 cm at 2π/5) were not needed: the Practice block draws its arc-length coverage from three answered exercises that need no figure instead. Omitted the decorative photograph of the 3rd-century Hierapolis water wheel, which carries no mathematics, and reworded the sentence that pointed at it. Converted the eleven “Try Its” into interactive fill-ins and multiple-choice questions with instant feedback; the 240° “sketch the angle” Try It became a quadrant-identification multiple choice, since the source asks only for a drawing with no separate checkable fact — the same adaptation used for the three “draw an angle” Practice items, each verified against the source’s own solution figure. Every coterminal-angle exercise states the representative it wants (“between $0^\circ$ and $360^\circ$”, “where $0\le\theta<2\pi$”), because a coterminal angle has infinitely many correct measures and only the stated one is graded. Every one of them remains a fill-in, including the two whose printed angle exceeds a full turn ($870^\circ$, and $\tfrac{44\pi}{9}$ in the Practice block): the grader reads a degree mark as the quantity it is rather than folding it onto one turn, so retyping the printed $870^\circ$ against the keyed $150^\circ$ is graded incorrect — confirmed against the real grader for all three spellings of the symbol. Adapted fourteen selected end-of-section exercises — three quadrant identifications, three degree/radian conversions, three coterminal-angle findings, three arc-length computations, a four-part angular/linear-speed problem, and a linear-speed problem — into sixteen interactive components in a closing Practice block, one group per objective.