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Drawing Angles in Standard Position

Properly defining an angle first requires that we define a ray. A ray is a directed line segment. It 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) can be named as ray EF, or in symbol form[latex]\,\stackrel{⟶}{EF}.[/latex]

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

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

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 arrow close to the vertex as in (Figure).

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[latex]\,360\,[/latex]degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol[latex]°.\,[/latex]For example,[latex]\,90\text{ degrees}=90°.[/latex]

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 (Figure).

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[latex]\,360°.\,[/latex]For example, to draw a[latex]\,90°\,[/latex]angle, we calculate that[latex]\,\frac{90°}{360°}=\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[latex]\,360°[/latex]angle, we calculate that[latex]\,\frac{360°}{360°}=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. See (Figure).

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[latex]\text{0°,}\,\text{90°,}\,\text{180°,}\,\text{270°,}[/latex]or[latex]\,\text{360°}.\,[/latex]See (Figure).

Quadrantal Angles

An angle is a quadrantal angle if its terminal side lies on an axis, including[latex]\text{0°,}\,\text{90°,}\,\text{180°,}\,\text{270°,}[/latex]or[latex]\,\text{360°}.[/latex]

How To

Given an angle measure in degrees, draw the angle in standard position.

1. Express the angle measure as a fraction of[latex]\,\text{360°}.[/latex]
2. Reduce the fraction to simplest form.
3. 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.

Drawing an Angle in Standard Position Measured in Degrees

1. Sketch an angle of[latex]\,30°\,[/latex]in standard position.
2. Sketch an angle of[latex]\,-135°\,[/latex]in standard position.

Show Solution

1. Divide the angle measure by[latex]\,360°.[/latex]

   [latex]\frac{30°}{360°}=\frac{1}{12}[/latex]

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

   [latex]\frac{1}{12}=\frac{1}{3}\left(\frac{1}{4}\right)[/latex]

   One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at[latex]\,30°,[/latex]as in (Figure).

   

2. Divide the angle measure by[latex]\,360°.[/latex]

   [latex]\frac{-135°}{360°}=-\frac{3}{8}[/latex]

   In this case, we can recognize that

   [latex]-\frac{3}{8}=-\frac{3}{2}\left(\frac{1}{4}\right)[/latex]

   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 (Figure).

   

Try It

Show an angle of[latex]\,240°\,[/latex]on a circle in standard position.

Show Solution

Converting Between Degrees and Radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.

The circumference of a circle is[latex]\,C=2\pi r.\,[/latex]If we divide both sides of this equation by[latex]\,r,[/latex]we create the ratio of the circumference, which is always[latex]\,2\pi ,[/latex]to the radius, regardless of the length of the radius. So the circumference of any circle is[latex]\,2\pi \approx 6.28\,[/latex]times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in (Figure).

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.

[latex]\begin{array}{ccc}\hfill 2\pi \text{ radians}& =& 360°\hfill \\ \hfill \pi \text{ radians}& =& \frac{360°}{2}=180°\hfill \\ \hfill 1\text{ radian}& =& \frac{180°}{\pi }\approx 57.3°\hfill \end{array}[/latex]

See (Figure). Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel.

Relating Arc Lengths to Radius

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

This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length[latex]\,s\,[/latex]to the radius r. See (Figure).

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

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

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

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

[latex]\begin{array}{ccc}\text{Smaller circle: }\frac{\frac{1}{2}\pi }{2}& =& \frac{1}{4}\pi \\ \text{Larger circle: }\frac{\frac{3}{4}\pi }{3}& =& \frac{1}{4}\pi \end{array}[/latex]

Since both ratios are[latex]\,\frac{1}{4}\pi ,[/latex]the angle measures of both circles are the same, even though the arc length and radius differ.

Radians

One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution[latex]\,\left(360°\right)\,[/latex]equals[latex]\,2\pi \,[/latex]radians. A half revolution[latex]\,\left(180°\right)\,[/latex]is equivalent to[latex]\,\pi \,[/latex]radians.

The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if[latex]\,s\,[/latex]is the length of an arc of a circle, and[latex]\,r\,[/latex]is the radius of the circle, then the central angle containing that arc measures[latex]\,\frac{s}{r}\,[/latex]radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.

A measure of 1 radian looks to be about[latex]\,60°.\,[/latex]Is that correct?

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

Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in (Figure), 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,[latex]\,360°.[/latex]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{array}{ccccc}\hfill \text{1 rotation}& =& 360°\hfill & =& 2\pi \text{ radians}\hfill \\ \hfill \frac{1}{2}\text{ rotation}& =& 180°\hfill & =& \pi \text{ radians}\hfill \\ \hfill \frac{1}{4}\text{ rotation}& =& 90°\hfill & =& \frac{\pi }{2}\text{ radians}\hfill \end{array}[/latex]

Identifying Special Angles Measured in Radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in (Figure). Memorizing these angles will be very useful as we study the properties associated with angles.

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

Finding a Radian Measure

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

Show Solution

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

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

So,

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

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

[latex]\begin{array}{ccc}\hfill \text{radian measure}& =& \frac{\frac{2\pi r}{3}}{r}\hfill \\ & =& \frac{2\pi r}{3r}\hfill \\ & =& \frac{2\pi }{3}\end{array}[/latex]

Try It

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

Show Solution

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