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Active Prelude to Calculus

Matthew Boelkins

Section 2.2 The Unit Circle

As demonstrated by several different examples in Section 2.1, certain periodic phenomena are closely linked to circles and circular motion. Rather than regularly work with circles of different center and radius, it turns out to be ideal to work with one standard circle and build all circular functions from it. The unit circle is the circle of radius \(1\) that is centered at the origin, \((0,0)\text{.}\)
If we pick any point \((x,y)\) that lies on the unit circle, the point is associated with a right triangle whose horizontal leg has length \(|x|\) and whose vertical leg has length \(|y|\text{,}\) as seen in Figure 2.2.1. By the Pythagorean Theorem, it follows that
\begin{equation*} x^2 + y^2 = 1\text{,} \end{equation*}
and this is the equation of the unit circle: a point \((x,y)\) lies on the unit circle if and only if \(x^2 + y^2 = 1\text{.}\)
Figure 2.2.1. Coordinates of a point on the unit circle.
Figure 2.2.2. A point traversing the unit circle.
To study the circular functions generated by the unit circle, we will also animate a point and let it traverse the circle. Starting at \((1,0)\) indicated by \(t_0\) in Figure 2.2.2, we see a sequence of points that result from traveling a distance along the circle that is \(1/24\) the circumference of the unit circle. Since the unit circle’s circumference is \(C = 2\pi r = 2\pi\text{,}\) it follows that the distance from \(t_0\) to \(t_1\) is
\begin{equation*} d = \frac{1}{24} \cdot 2\pi = \frac{\pi}{12}\text{.} \end{equation*}
As we work to better understand the unit circle, we will commonly use fractional multiples of \(\pi\) as these result in natural distances traveled along the unit circle.

Preview Activity 2.2.1.

In Figure 2.2.3 there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is \(2\pi\text{,}\) each of the points is \(\frac{1}{24} \cdot 2\pi = \frac{\pi}{12}\) units apart (traveled along the circle). Thus, the first point counterclockwise from \((1,0)\) corresponds to the distance \(t = \frac{\pi}{12}\) traveled along the unit circle. The second point is twice as far, and thus \(t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6}\) units along the circle away from \((1,0)\text{.}\)
Figure 2.2.3. The unit circle with \(24\) equally-spaced points.
  1. Label each of the subsequent points on the unit circle with the exact distance they lie counter-clockwise away from \((1,0)\text{;}\) write each fraction in lowest terms.
  2. Which distance along the unit circle corresponds to \(\frac{1}{4}\) of a full rotation around? to \(\frac{5}{8}\) of a full rotation?
  3. One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at \((0,0)\) in the unit circle, we say the angle’s measure is \(1\) radian provided that the angle intercepts an arc of the circle that is \(1\) unit in length, as pictured in Figure 2.2.4. Note particularly that an angle measuring \(1\) radian intercepts an arc of the same length as the circle’s radius.
    Figure 2.2.4. An angle \(\theta\) of measure \(1\) radian.
    Suppose that \(\alpha\) and \(\beta\) are each central angles and that their respective radian measures are \(\alpha = \frac{\pi}{3}\) and \(\beta = \frac{3\pi}{4}\text{.}\) Sketch the angles \(\alpha\) and \(\beta\) on the unit circle in Figure 2.2.3.
  4. What is the radian measure that corresonds to a \(90^\circ\) angle?

Subsection 2.2.1 Radians and degrees

In Preview Activity 2.2.1, we introduced the idea of radian measure of an angle. Here we state the formal definition of this term.

Definition 2.2.5.

An angle whose vertex is at the center of a circle 1  measures \(1\) radian provided that the arc the angle intercepts on the circle equals the radius of the circle.
As seen in Figure 2.2.4, in the unit circle this means that a central angle has measure \(1\) radian whenever it intercepts an arc of length \(1\) unit along the circumference. Because of this important correspondence between the unit circle and radian measure (one unit of arc length on the unit circle corresponds to \(1\) radian), we focus our discussion of radian measure within the unit circle.
Since there are \(2\pi\) units of length along the unit circle’s circumference it follows there are \(\frac{1}{4} \cdot 2\pi = \frac{\pi}{2}\) units of length in \(\frac{1}{4}\) of a revolution. We also know that \(\frac{1}{4}\) of a revolution corresponds to a central angle that is a right angle, whose familiar degree measure is \(90^\circ\text{.}\) If we extend to a central angle that intercepts half the circle, we see similarly that \(\pi\) radians corresponds to \(180^\circ\text{;}\) this relationship enables us to convert angle measures from radians to degrees and vice versa.

Converting between radians and degrees.

An angle whose radian measure is \(1\) radian has degree measure \(\frac{180}{\pi} ^\circ\text{.}\) An angle whose degree measure is \(1^\circ\) has radian measure \(\frac{\pi}{180}\text{.}\)

Activity 2.2.2.

Convert each of the following quantities to the alternative measure: degrees to radians or radians to degrees.
  1. \(\displaystyle 30^\circ\)
  2. \(\frac{2\pi}{3}\) radians
  3. \(\frac{5\pi}{4}\) radians
  4. \(\displaystyle 240^\circ\)
  5. \(\displaystyle 17^\circ\)
  6. \(2\) radians
Note that in Figure 2.2.3 in the Preview Activity, we labeled \(24\) equally spaced points with their respective distances around the unit circle counterclockwise from \((1,0)\text{.}\) Because these distances are on the unit circle, they also correspond to the radian measure of the central angles that intercept them. In particular, each central angle with one of its sides on the positive \(x\)-axis generates a unique point on the unit circle, and with it, an associated length intercepted along the circumference of the circle. A good exercise at this point is to return to Figure 2.2.3 and label each of the noted points with the degree measure that is intercepted by a central angle with one side on the positive \(x\)-axis, in addition to the arc lengths (radian measures) already identified.

Subsection 2.2.2 Special points on the unit circle

Our in-depth study of the unit circle is motivated by our desire to better understand the behavior of circular functions. Recall that as we traverse a circle, the height of the point moving along the circle generates a function that depends on distance traveled along the circle. Wherever possible, we’d like to be able to identify the exact height of a given point on the unit circle. Two special right triangles enable us to locate exactly an important collection of points on the unit circle.

Activity 2.2.3.

In what follows, we work to understand key relationships in \(45^\circ\)-\(45^\circ\)-\(90^\circ\) and \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangles.
Figure 2.2.6. A right triangle with two \(45^\circ\) angles.
Figure 2.2.7. A right triangle with a \(30^\circ\) angle.
  1. For the \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with legs \(x\) and \(y\) and hypotenuse \(1\text{,}\) what does the fact that the triangle is isosceles tell us about the relationship between \(x\) and \(y\text{?}\) What are their exact values?
  2. Now consider the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse \(1\) and the longer leg lying along the positive \(x\)-axis. What special kind of triangle is formed when we reflect this triangle across the \(x\)-axis? How can we use this perspective to determine the exact values of \(x\) and \(y\text{?}\)
  3. Suppose we consider the related \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse \(1\) and the shorter leg lying along the positive \(x\)-axis. What are the exact values of \(x\) and \(y\) in this triangle?
  4. We know from the conversion factor from degrees to radians that an angle of \(30^\circ\) corresponds to an angle measuring \(\frac{\pi}{6}\) radians, an angle of \(45^\circ\) corresponds to \(\frac{\pi}{4}\) radians, and \(60^\circ\) corresponds to \(\frac{\pi}{3}\) radians.
    Figure 2.2.8. An angle measuring \(\frac{\pi}{6}\) radians.
    Figure 2.2.9. An angle measuring \(\frac{\pi}{4}\) radians.
    Figure 2.2.10. An angle measuring \(\frac{\pi}{3}\) radians.
    Use your work in (a), (b), and (c) to label the noted point in each of Figure 2.2.8, Figure 2.2.9, and Figure 2.2.10, respectively, with its exact coordinates.
Our work in Activity 2.2.3 enables us to identify exactly the location of \(12\) special points on the unit circle. In part (d) of the activity, we located the three noted points in Figure 2.2.11 along with their respective radian measures. By symmetry across the coordinate axes and thinking about the signs of coordinates in the other three quadrants, we can now identify all of the coordinates of the remaining \(9\) points.
Figure 2.2.11. The unit circle with \(16\) special points whose location we can determine exactly.
In addition, we note that there are four additional points on the circle that we can locate exactly: the four points that correspond to angle measures of \(0\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\pi\text{,}\) and \(\frac{3\pi}{2}\) radians, which lie where the coordinate axes intersect the circle. Each such point has \(0\) for one coordinate and \(\pm 1\) for the other. Labeling all of the remaining points in Figure 2.2.11 is an important exercise that you should do on your own.
Finally, we note that we can identify any point on the unit circle exactly simply by choosing one of its coordinates. Since every point \((x,y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\text{,}\) if we know the value of \(x\) or \(y\) and the quadrant in which the point lies, we can determine the other coordinate exactly.

Subsection 2.2.3 Special points and arc length in non-unit circles

All of our work with the unit circle can be extended to circles centered at the origin with different radii, since a circle with a larger or smaller radius is a scaled version of the unit circle. For instance, if we instead consider a circle of radius \(7\text{,}\) the coordinates of every point on the unit circle are magnified by a factor of \(7\text{,}\) so the point that corresponds to an angle such as \(\theta = \frac{2\pi}{3}\) has coordinates \(\left( -\frac{7}{2}, \frac{7\sqrt{3}}{2} \right)\text{.}\) Distance along the circle is magnified by the same factor: the arc length along the unit circle from \((0,0)\) to \(\left( -\frac{7}{2}, \frac{7\sqrt{3}}{2} \right)\) is \(7 \cdot \frac{2\pi}{3}\text{,}\) since the arc length along the unit circle for this angle is \(\frac{2\pi}{3}\text{.}\)
If we think more generally about a circle of radius \(r\) with a central angle \(\theta\) that intercepts an arc of length \(s\text{,}\) we see how the magnification factor \(r\) (in comparison to the unit circle) connects arc length and the central angle according to the following principle.

Connecting arc length and angles in non-unit circles.

If a central angle measuring \(\theta\) radians intercepts an arc of length \(s\) in a circle of radius \(r\text{,}\) then
\begin{equation*} s = r \theta\text{.} \end{equation*}
In the unit circle, where \(r = 1\text{,}\) the equation \(s = r\theta\) demonstrates the familiar fact that arc length matches the radian measure of the central angle. Moreover, we also see how this formula aligns with the definition of radian measure: if the arc length and radius are equal, then the angle measures \(1\) radian.

Activity 2.2.4.

Determine each of the following values or points exactly.
  1. In a circle of radius \(11\text{,}\) the arc length intercepted by a central angle of \(\frac{5\pi}{3}\text{.}\)
  2. In a circle of radius \(3\text{,}\) the central angle measure that intercepts an arc of length \(\frac{\pi}{4}\text{.}\)
  3. The radius of the circle in which an angle of \(\frac{7\pi}{6}\) intercepts an arc of length \(\frac{\pi}{2}\text{.}\)
  4. The exact coordinates of the point on the circle of radius \(5\) that lies \(\frac{25\pi}{6}\) units counterclockwise along the circle from \((5,0)\text{.}\)

Subsection 2.2.4 Summary

  • The radian measure of an angle connects the measure of a central angle in a circle to the radius of the circle. A central angle has radian measure \(1\) provided that it intercepts an arc of length equal to the circle’s radius. In the unit circle, a central angle’s radian measure is precisely the same numerical value as the length of the arc it intercepts along the circle.
  • If we begin at the point \((1,0)\) and move counterclockwise along the unit circle, there are natural special points on the unit circle that correspond to angles of measure \(30^\circ\text{,}\) \(45^\circ\text{,}\) \(60^\circ\text{,}\) and their multiples. We can count in \(30^\circ\) increments and identify special points that correspond to angles of measure \(30^\circ\text{,}\) \(60^\circ\text{,}\) \(90^\circ\text{,}\) \(120^\circ\text{,}\) and so on; doing likewise with \(45^\circ\text{,}\) these correspond to angles of \(45^\circ\text{,}\) \(90^\circ\text{,}\) \(135^\circ\text{,}\) etc. In radian measure, these sequences together give us the important angles \(\frac{\pi}{6}\text{,}\) \(\frac{\pi}{4}\text{,}\) \(\frac{\pi}{3}\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\frac{2\pi}{3}\text{,}\) \(\frac{3\pi}{4}\text{,}\) and so on. Together with our work involving \(45^\circ\)-\(45^\circ\)-\(90^\circ\) and \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangles in Activity 2.2.3, we are able to identify the exact locations of all of the points in Figure 2.2.11.
  • In any circle of radius \(r\text{,}\) if a central angle of measure \(\theta\) radians intercepts an arc of length \(s\text{,}\) then it follows that
    \begin{equation*} s = r\theta\text{.} \end{equation*}
    This shows that arc length, \(s\text{,}\) is magnified along with the size of the radius, \(r\text{,}\) of the circle.

Exercises 2.2.5 Exercises

1.

Find \(t\) for the following terminal points:
(a) \(P(0,1)\text{,}\) \(t=\) ;
(b) \(P(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\text{,}\) \(t=\) ;
(c) \(P(-1,0)\text{,}\) \(t=\) ;
(d) \(P(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\text{,}\) \(t=\) ;
(e) \(P(0,-1)\text{,}\) \(t=\) ;
(f) \(P(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\text{,}\) \(t=\) .

2.

What is the exact radian angle measure for \(45^{\circ}\) as a fraction of \(\pi\text{?}\) radians
What is a decimal approximation for the radian angle measure for \(45^{\circ}\) accurate to three decimal places? radians

3.

What is the exact radian angle measure for \(22^{\circ}\) as a fraction of \(\pi\text{?}\) radians
What is a decimal approximation for the radian angle measure for \(22^{\circ}\) accurate to three decimal places? radians

4.

Find the length of an arc on a circle of radius \(6\) corresponding to an angle of \(90^{\circ}\text{.}\)
Arc length = units. (Give an exact answer as a fraction, not a decimal approximation.)

5.

What is the length of an arc cut off by an angle of 2.5 radians on a circle of radius 4 inches?
(include units: )

6.

What angle (in degrees) corresponds to 17.7 rotations around the unit circle?
17.7 rotations is an angle of degrees.

7.

An angle of \(\displaystyle \frac{ 5 \pi}{6}\) radians can be converted to an angle of degrees.

8.

Let \((x,y)\) by a point on the unit circle. In each of the following situations, determine requested value exactly.
  1. Suppose that \(x = -0.3\) and \(y\) is negative. Find the value of \(y\text{.}\)
  2. Suppose that \((x,y)\) lies in Quadrant II and \(x = -2y\text{.}\) Find the values of \(x\) and \(y\text{.}\)
  3. Suppose that \((x,y)\) lies a distance of \(\frac{29\pi}{6}\) units clockwise around the circle from \((1,0)\text{.}\) Find the values of \(x\) and \(y\text{.}\)
  4. At what exact point(s) does the line \(y = \frac{1}{2}x + \frac{1}{2}\) intersect the unit circle?

9.

The unit circle is centered at \((0,0)\) and radius \(r = 1\text{,}\) from which the Pythagorean Theorem tells us that any point \((x,y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\text{.}\)
  1. Explain why any point \((x,y)\) on a circle of radius \(r\) centered at \((h,k)\) satisfies the equation \((x-h)^2 + (y-k)^2 = r^2\text{.}\)
  2. Determine the equation of a circle centered at \((-3,5)\) with radius \(r = 2\text{.}\)
  3. Suppose that the unit circle is magnified by a factor of \(5\) and then shifted \(4\) units right and \(7\) units down. What is the equation of the resulting circle?
  4. What is the length of the arc intercepted by a central angle of \(\frac{2\pi}{3}\) radians in the circle \((x-1)^2 + (y-3)^2 = 16\text{?}\)
  5. Suppose that the line segment from \((-2,-1)\) to \((4,2)\) is a diameter of a circle. What is the circle’s center, radius, and equation?

10.

Consider the circle whose center is \((0,0)\) and whose radius is \(r = 5\text{.}\) Let a point \((x,y)\) traverse the circle counterclockwise from \((5,0)\text{,}\) and say the distance along the circle from \((5,0)\) is represented by \(d\text{.}\)
  1. Consider the point \((a,b)\) that is generated by the central angle \(\theta\) with vertices \((5,0)\text{,}\) \((0,0)\text{,}\) and \((a,b)\text{.}\) If \(\theta = \frac{\pi}{6}\text{,}\) what are the exact values of \(a\) and \(b\text{?}\)
  2. Answer the same question as in (a) except with \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{\pi}{3}\text{.}\)
  3. How far has the point \((x,y)\) traveled after it has traversed the circle one full revolution?
  4. Let \(h = f(d)\) be the circular function that tracks the height of the point \((x,y)\) as a function of distance, \(d\text{,}\) traversed counterclockwise from \((5,0)\text{.}\) Sketch an accurate graph of \(f\) through two full periods, labeling several special points on the graph as well as the horizontal and vertical scale of the axes.
We often call such an angle a central angle.
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