In Section AppAngles, we review the concept of (oriented) angles and degree measure. While degrees are the unit of choice for many applications of trigonometry, we introduce here the concept of the radian measure of an angle. As we will see, this concept naturally ties angles to real numbers. While the concept may seem foreign at first, we assure the reader that the utility of radian measure in modeling real-world phenomena is well worth the effort. We begin our development with a definition from Geometry.

While Definition pidefn is quite possibly the ‘standard’ definition of \(\pi \), the authors would be remiss if we didn’t mention that buried in this definition is actually a theorem. As the reader is probably aware, the number \(\pi \) is a mathematical constant - that is, it doesn’t matter which circle is selected, the ratio of its circumference to its diameter will have the same value as any other circle. While this is indeed true, it is far from obvious and leads to a counterintuitive scenario which is explored in the Exercises. Since the diameter of a circle is twice its radius, we can quickly rearrange the equation in Definition pidefn to get a formula more useful for our purposes, namely: \(2 \pi = \frac {C}{r}\). Hence, for any circle, the ratio of its circumference to its radius is \(2\pi \).

Suppose we take a portion of a circle and compare some arc measuring \(s\) units in length along the circumference to the radius, \(r\). Let \(\theta \) be the central angle subtended by this arc, that is, an angle whose vertex is the center of the circle and whose determining rays pass through the endpoints of the arc.

Using the GeoGebra interactive below, click and drag the point \(P\) to change the length of the arc, \(s\) (and hence the angle, \(\theta \).) If we keep \(\theta \) the same and adjust the radius of the circle, \(r\), notice that the ratio \(\frac {s}{r}\) remains constant. ^[1] (Also note that as we stretch \(s\) to cover one circumference (hence one complete revolution for \(\theta \)) the ratio approaches \(2\pi \).)

The same proportionality (similarity) arguments used to show \(\pi \) is a constant shows the ratio \(\frac {s}{r}\) is also constant among all circles. It is this ratio, \(\frac {s}{r}\), which defines the radian measure of an angle.

To get a better feel for radian measure, consider the GeoGebra interactive below. Note there are two sliders: one to adjust the radian measure of the central angle, \(\theta \), and one to adjust the radius, \(r\). The length of the subtended arc, \(s\), is marked off into increments of \(r\). Notice, for instance, that for an angle \(\theta \) with radian measure \(2\), the length of \(s\) is exactly double the radius, \(r\). That is, \(s\) sweeps out two ‘radius lengths’ worth of circumference. Likewise, if \(\theta = 3\) radians, every subtended arc sweeps out three radius lengths.

In general, the radian measure of an angle \(\theta \) tells us how many ‘radius lengths’ we need to sweep out along the circle to subtend the angle \(\theta \).

Since one revolution sweeps out the circumference \(2\pi r\), one revolution has radian measure \(\frac {2 \pi r}{r} = 2 \pi \). From this we can find the radian measure of other central angles using proportions, just like we did with degrees. For instance, half of a revolution has radian measure \(\frac {1}{2} (2 \pi ) = \pi \), a quarter revolution has radian measure \(\frac {1}{4} (2 \pi ) = \frac {\pi }{2}\), and so forth. Note that, by definition, the radian measure of an angle is a length divided by another length so that these measurements are actually dimensionless and are considered ‘pure’ numbers. For this reason, we do not use any symbols to denote radian measure, but we use the word ‘radians’ to denote these dimensionless units as needed. For instance, we say one revolution measures ‘\(2\pi \) radians,’ half of a revolution measures ‘\(\pi \) radians,’ and so forth.

As with degree measure, the distinction between the angle itself and its measure is often blurred in practice, so when we write ‘\(\theta = \frac {\pi }{2}\)’, we mean \(\theta \) is an angle which measures \(\frac {\pi }{2}\) radians. ^[2] (The authors are well aware that we are now identifying radians with real numbers. We will justify this shortly.) We extend radian measure to oriented angles, just as we did with degrees beforehand, so that a positive measure indicates counter-clockwise rotation and a negative measure indicates clockwise rotation. ^[3] (This, in turn, endows the subtended arcs with an orientation as well. We address this in short order.) Much like before, two positive angles \(\alpha \) and \(\beta \) are supplementary if \(\alpha + \beta = \pi \) and complementary if \(\alpha + \beta = \frac {\pi }{2}\). Finally, we leave it to the reader to show that when using radian measure, two angles \(\alpha \) and \(\beta \) are coterminal if and only if \(\beta = \alpha + 2\pi k\) for some integer \(k\).

It is worth mentioning that we could have plotted the angles in Example orientedcoterminalradian by first converting them to degree measure and following the procedure set forth in Example orientedcoterminaldegree. While converting back and forth from degrees and radians is certainly a good skill to have, it is best that you learn to ‘think in radians’ as well as you can ‘think in degrees’. The authors would, however, be derelict in our duties if we ignored the basic conversion between these systems altogether.

Since one revolution counter-clockwise measures \(360^{\circ }\) and the same angle measures \(2 \pi \) radians, we can use the proportion \(\frac {2 \pi \, \text {radians}}{360^{\circ }}\), or its reduced equivalent, \(\frac {\pi \, \text {radians}}{180^{\circ }}\), as the conversion factor between the two systems. For example, to convert \(60^{\circ }\) to radians we find \(60^{\circ } \left ( \frac {\pi \, \text {radians}}{180^{\circ }}\right ) = \frac {\pi }{3} \, \text {radians}\), or simply \(\frac {\pi }{3}\). To convert from radian measure back to degrees, we multiply by the ratio \(\frac {180^{\circ }}{\pi \, \text {radian}}\). For example, \(-\frac {5 \pi }{6} \, \text {radians}\) is equal to \(\left (-\frac {5 \pi }{6} \, \text {radians} \right ) \left ( \frac {180^{\circ }}{\pi \, \text {radians}}\right ) = -150^{\circ }\). ^[4] (Note that the negative sign indicates clockwise rotation in both systems, and so it is carried along accordingly.) Hence, an angle which measures \(1\) in radian measure is equal to \(\frac {180^{\circ }}{\pi } \approx 57.2958^{\circ }\). To summarize:

In light of Example orientedcoterminalradian and Equation degreenradianconversion, the reader may well wonder what the allure of radian measure is. The numbers involved are, admittedly, much more complicated than degree measure. The answer lies in how easily angles in radian measure can be identified with real numbers.

Consider the Unit Circle, \(x^2 + y^2 = 1\), as drawn below, the angle \(\theta \) in standard position and the corresponding arc measuring \(s\) units in length. By definition, and the fact that the Unit Circle has radius 1, the radian measure of \(\theta \) is \(\frac {s}{r}=\frac {s}{1} = s\) so that, once again blurring the distinction between an angle and its measure, we have \(\theta = s\).

In order to identify real numbers with oriented angles, we essentially ‘wrap’ the real number line around the Unit Circle and associating to each real number \(t\) an oriented arc on the Unit Circle with initial point \((1,0)\).

Viewing the vertical line \(x=1\) as another real number line demarcated like the \(y\)-axis, given a real number \(t>0\), we ‘wrap’ the (vertical) interval \([0,t]\) around the Unit Circle in a counter-clockwise fashion.

This process is best seen in the GeoGebra interactive below on the left. By adjusting the slider upwards from \(0\), we see as we move up the vertical line \(x = 1\) from \((1,0)\) to the point \((1,t)\), an arc of length \(t\) units is traced out counter-clockwise along the Unit Circle. Since this arc has a length of \(t\) units, the corresponding angle has radian measure equal to \(t\).

If \(t<0\), we wrap the interval \([t,0]\) clockwise around the Unit Circle. This is seen above using the interactive on the right. Adjusting the slider and moving downwards, we once again trace out an interval on the line \(x = 1\) from \((1,0)\) to \((1,t)\). Here, the arc wraps clockwise around the Unit Circle. Since we have defined clockwise rotation as having negative radian measure, the angle determined by this arc has radian measure equal to \(t\) in this case as well.

Finally, note that if \(t=0\), we are at the point \((1,0)\) on the \(x\)-axis which corresponds to an angle with radian measure \(0\). In this way, we identify each real number \(t\) with the corresponding angle with radian measure \(t\).

1 Applications of Radian Measure: Circular Motion

Now that we have paired angles with real numbers via radian measure, a whole world of applications awaits us. Our first excursion into this realm comes by way of circular motion. Suppose an object is moving as pictured below along a circular path of radius \(r\) from the point \(P\) to the point \(Q\) in an amount of time \(t\).

Here \(s\) represents a displacement so that \(s > 0\) means the object is traveling in a counter-clockwise direction and \(s<0\) indicates movement in a clockwise direction. Note that with this convention the formula we used to define radian measure, namely \(\theta = \frac {s}{r}\), still holds since a negative value of \(s\) incurred from a clockwise displacement matches the negative we assign to \(\theta \) for a clockwise rotation. In Physics, the average velocity of the object, denoted \(\overline {v}\) and read as ‘\(v\)-bar’, is defined as the average rate of change of the position of the object with respect to time. ^[5] (See Definition ?? in Section ?? for a review of this concept.) As a result, we have \(\overline {v} = \frac {\text {displacement}}{\text {time}} = \frac {s}{t}\). The quantity \(\overline {v}\) has units of \(\frac {\text {length}}{\text {time}}\) and conveys two ideas: the direction in which the object is moving and how fast the position of the object is changing. The contribution of direction in the quantity \(\overline {v}\) is either to make it positive (in the case of counter-clockwise motion) or negative (in the case of clockwise motion), so that the quantity \(\left | \overline {v} \right |\) quantifies how fast the object is moving - it is the speed of the object. Measuring \(\theta \) in radians we have \(\theta = \frac {s}{r}\) thus \(s = r \theta \) and

The quantity \(\frac {\theta }{t}\) is called the average angular velocity of the object. It is denoted by \(\overline {\omega }\) and is read ‘omega-bar’. The quantity \(\overline {\omega }\) is the average rate of change of the angle \(\theta \) with respect to time and thus has units \(\frac {\text {radians}}{\text {time}}\). If \(\overline {\omega }\) is constant throughout the duration of the motion, then it can be shown ^[6] (You guessed it, using Calculus …) that the average velocities involved, namely \(\overline {v}\) and \(\overline {\omega }\), are the same as their instantaneous counterparts, \(v\) and \(\omega \), respectively. In this case, \(v\) is simply called the ‘velocity’ of the object and \(\omega \) is called the ‘angular velocity.’ ^[7] (See Example ?? in Section ?? for more of a discussion on instantaneous velocity.)

If the path of the object were ‘uncurled’ from a circle to form a line segment, then the velocity of the object on that line segment would be the same as the velocity on the circle. For this reason, the quantity \(v\) is often called the linear velocity of the object in order to distinguish it from the angular velocity, \(\omega \). Putting together the ideas of the previous paragraph, we get the following.

We need to talk about units here. The units of \(v\) are \(\frac {\text {length}}{\text {time}}\), the units of \(r\) are length only, and the units of \(\omega \) are \(\frac {\text {radians}}{\text {time}}\). Thus the left hand side of the equation \(v = r \omega \) has units \(\frac {\text {length}}{\text {time}}\), whereas the right hand side has units \(\text {length} \cdot \frac {\text {radians}}{\text {time}} = \frac {\text {length} \cdot \text {radians}}{\text {time}} \). The supposed contradiction in units is resolved by remembering that radians are a dimensionless quantity and angles in radian measure are identified with real numbers so that the units \(\frac {\text {length} \cdot \text {radians}}{\text {time}}\) reduce to the units \(\frac {\text {length}}{\text {time}}\). We are long overdue for an example.

It is worth noting that the quantity \(\frac {1 \, \text {revolution}}{24 \, \text {hours}}\) in Example 3 is called the ordinary frequency of the motion and is usually denoted by the variable \(f\). The ordinary frequency is a measure of how often an object makes a complete cycle of the motion. The fact that \(\omega = 2\pi f\) suggests that \(\omega \) is also a frequency. Indeed, it is called the angular frequency of the motion. On a related note, the quantity \(T = \frac {1}{f}\) is called the period of the motion and is the amount of time it takes for the object to complete one cycle of the motion. In the scenario of Example 3, the period of the motion is 24 hours, or one day.

The concepts of frequency and period help frame the equation \(v = r \omega \) in a new light. That is, if \(\omega \) is fixed, points which are farther from the center of rotation need to travel faster to maintain the same angular frequency since they have farther to travel to make one revolution in one period’s time. The distance of the object to the center of rotation is the radius of the circle, \(r\), and is the ‘magnification factor’ which relates \(\omega \) and \(v\). We will have more to say about frequencies and periods in Section ??. While we have exhaustively discussed velocities associated with circular motion, we have yet to discuss a more natural question: if an object is moving on a circular path of radius \(r\) with a fixed angular velocity (frequency) \(\omega \), what is the position of the object at time \(t\)? The answer to this question is the very heart of Trigonometry and is answered in the next section.