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Precalculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

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Section 6.5 The Unit Circle (TR5)

Subsection 6.5.1 Activities

Remark 6.5.1.

In Section 6.4, we learned how to find the exact values of the six trigonometric ratios for the special acute angles \(30^\circ\text{,}\) \(45^\circ\text{,}\) and \(60^\circ\text{.}\) In this section, we will use that knowledge and expand to finding the exact trig values of any multiple of those angles.

Definition 6.5.2.

The unit circle is the circle of radius \(1\) centered at the origin on the coordinate plane.
Unit circle
Figure 6.5.3.

Activity 6.5.4.

Let \(\theta\) be the angle shown below in standard form. Notice that the terminal side intersects with the unit circle. (Note: We will assume a circle drawn in the this context is the unit circle unless told otherwise.) We will label that point of intersection as \((x,y)\text{.}\)
(a)
What is the length of line segment \(r\text{,}\) whose endpoints are the origin and the point \((x,y)\) ?
  1. \(\displaystyle 1\)
  2. \(\displaystyle 2\)
  3. \(\displaystyle 3\)
  4. cannot be determined
Answer.
A
(b)
We will now create a right triangle using the previous line segment \(r\) as the hypotenuse. Draw in a line segment of length \(x\) and another of length \(y\) to create such a triangle.
Answer.
(c)
Using the triangle you’ve just created, find \(\cos \theta\text{.}\)
  1. \(\displaystyle \dfrac{x}{y}\)
  2. \(\displaystyle \dfrac{1}{x}\)
  3. \(\displaystyle \dfrac{x}{1}\)
  4. \(\displaystyle \dfrac{1}{y}\)
  5. \(\displaystyle \dfrac{y}{1}\)
Answer.
C
(d)
Using that same triangle, find \(\sin \theta\text{.}\)
  1. \(\displaystyle \dfrac{x}{y}\)
  2. \(\displaystyle \dfrac{1}{x}\)
  3. \(\displaystyle \dfrac{x}{1}\)
  4. \(\displaystyle \dfrac{1}{y}\)
  5. \(\displaystyle \dfrac{y}{1}\)
Answer.
E
(e)
Solve for \(x\) in one of the equations you’ve found above to determine an expression for the \(x\)-value of the point \((x,y)\) .
  1. \(\displaystyle y\cos \theta\)
  2. \(\displaystyle y\sin \theta\)
  3. \(\displaystyle \cos \theta\)
  4. \(\displaystyle \sin \theta\)
  5. \(\displaystyle \dfrac{1}{\cos \theta}\)
  6. \(\displaystyle \dfrac{1}{\sin \theta}\)
Answer.
C
(f)
Solve for \(y\) in one of the equations you’ve found above to determine an expression for the \(y\)-value of the point \((x,y)\) .
  1. \(\displaystyle y\cos \theta\)
  2. \(\displaystyle y\sin \theta\)
  3. \(\displaystyle \cos \theta\)
  4. \(\displaystyle \sin \theta\)
  5. \(\displaystyle \dfrac{1}{\cos \theta}\)
  6. \(\displaystyle \dfrac{1}{\sin \theta}\)
Answer.
D

Remark 6.5.5.

From the previous activity, we have found a connection between the sine and cosine values of an angle \(\theta\) and the coordinates \((x,y)\) of the point at which that angle intersects the unit circle. Namely,
\begin{equation*} x=\cos \theta \, \text{ and }\, y = \sin \theta \end{equation*}

Activity 6.5.6.

Consider each angle \(\theta\) given below. Find the coordinates \((x,y)\) for the point at which \(\theta\) intersects the unit circle.
(a)
\(\theta = \dfrac{\pi}{4}\)
  1. \(\displaystyle \left( \dfrac{1}{2}, \dfrac{\sqrt{3}}{2} \right)\)
  2. \(\displaystyle \left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right)\)
  3. \(\displaystyle \left( \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} \right)\)
  4. \(\displaystyle (0,1)\)
  5. \(\displaystyle (1,0)\)
Answer.
B
(b)
\(\theta = 30^\circ\)
  1. \(\displaystyle \left( \dfrac{1}{2}, \dfrac{\sqrt{3}}{2} \right)\)
  2. \(\displaystyle \left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right)\)
  3. \(\displaystyle \left( \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} \right)\)
  4. \(\displaystyle (0,1)\)
  5. \(\displaystyle (1,0)\)
Answer.
B
(c)
\(\theta = \dfrac{\pi}{3}\)
  1. \(\displaystyle \left( \dfrac{1}{2}, \dfrac{\sqrt{3}}{2} \right)\)
  2. \(\displaystyle \left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right)\)
  3. \(\displaystyle \left( \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} \right)\)
  4. \(\displaystyle (0,1)\)
  5. \(\displaystyle (1,0)\)
Answer.
A
(d)
\(\theta = 0^\circ\)
  1. \(\displaystyle \left( \dfrac{1}{2}, \dfrac{\sqrt{3}}{2} \right)\)
  2. \(\displaystyle \left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right)\)
  3. \(\displaystyle \left( \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} \right)\)
  4. \(\displaystyle (0,1)\)
  5. \(\displaystyle (1,0)\)
Answer.
E
(e)
\(\theta = \dfrac{\pi}{2}\)
  1. \(\displaystyle \left( \dfrac{1}{2}, \dfrac{\sqrt{3}}{2} \right)\)
  2. \(\displaystyle \left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right)\)
  3. \(\displaystyle \left( \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} \right)\)
  4. \(\displaystyle (0,1)\)
  5. \(\displaystyle (1,0)\)
Answer.
D

Remark 6.5.7.

In Activity 6.5.6, we found \((x,y)\)-coordinates (and thus the sine and cosine values) for angles that terminated either in Quadrant 1 or on an axis adjacent to Quadrant 1. We’ll now expand to angles that terminate elsewhere, using our knowledge of the cosine and sine values of angles in the first quadrant along with how reflections over the \(x\) and \(y\) axes affect the signs of the coordinates. (See Section 2.4 for a reminder on how these reflections work.)

Activity 6.5.8.

Let’s consider the angle \(\theta=150^\circ\text{,}\) drawn below with the unit circle.
(a)
The angle below represents the reference angle for \(\theta=150^\circ\text{,}\) which is the smallest angle between the terminal side of \(\theta\) and the \(x\)-axis. What is the measure of this reference angle?
  1. \(\displaystyle 0^\circ\)
  2. \(\displaystyle 30^\circ\)
  3. \(\displaystyle 45^\circ\)
  4. \(\displaystyle 60^\circ\)
  5. \(\displaystyle 75^\circ\)
Answer.
B
(b)
We can find the sine and cosines values of our original angle, \(\theta=150^\circ\text{,}\) by using the reference angle to help. Find the point \((x_1,y_1)\text{,}\) where the terminal side of the \(30^\circ\) angle intersects the unit circle.
  1. \(\displaystyle \left(\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)\)
  2. \(\displaystyle \left(\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)\)
  3. \(\displaystyle \left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)\)
  4. \(\displaystyle \left(\dfrac{\sqrt{3}}{2},-\dfrac{1}{2}\right)\)
  5. \(\displaystyle \left(\dfrac{-\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)\)
Answer.
B
(c)
How does the point you’ve just found compare with the point \((x,y)\text{,}\) where the terminal edge of \(\theta=150^\circ\) intersects the unit circle?
  1. The \(x\)-values and the \(y\)-values are switched with each other.
  2. The \(x\)-values will be the same, but the \(y\)-values will have opposite signs.
  3. The \(y\)-values will be the same, but the \(x\)-values will have opposite signs.
  4. The \(x\)-values and the \(y\)-values will both have opposite signs.
Answer.
C
(d)
What are the sine and cosine of \(\theta=150^\circ\text{?}\)
  1. \(\sin 150^\circ = \dfrac{1}{2}\) and \(\cos 150^\circ = \dfrac{\sqrt{3}}{2}\)
  2. \(\sin 150^\circ = \dfrac{\sqrt{3}}{2}\) and \(\cos 150^\circ = \dfrac{1}{2}\)
  3. \(\sin 150^\circ = -\dfrac{\sqrt{3}}{2}\) and \(\cos 150^\circ = \dfrac{1}{2}\)
  4. \(\sin 150^\circ = \dfrac{1}{2}\) and \(\cos 150^\circ = -\dfrac{\sqrt{3}}{2}\)
  5. \(\sin 150^\circ = -\dfrac{1}{2}\) and \(\cos 150^\circ = \dfrac{\sqrt{3}}{2}\)
Answer.
D

Activity 6.5.9.

Let’s consider the angle \(\theta=\dfrac{4\pi}{3}\text{,}\) drawn below with the unit circle.
(a)
The angle below represents the reference angle for \(\theta=\dfrac{4\pi}{3}\text{,}\) which is the smallest angle between the terminal side of \(\theta\) and the \(x\)-axis. What is the measure of this reference angle?
  1. \(\displaystyle \dfrac{\pi}{2}\)
  2. \(\displaystyle \dfrac{\pi}{3}\)
  3. \(\displaystyle \dfrac{\pi}{4}\)
  4. \(\displaystyle \dfrac{\pi}{5}\)
  5. \(\displaystyle \dfrac{\pi}{6}\)
Answer.
B
(b)
We can find the sine and cosines values of our original angle, \(\theta=\dfrac{4\pi}{3}\text{,}\) by using the reference angle to help. Find the point \((x_1,y_1)\text{,}\) where the terminal side of the angle \(\dfrac{\pi}{3}\) intersects the unit circle.
  1. \(\displaystyle \left(\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)\)
  2. \(\displaystyle \left(\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)\)
  3. \(\displaystyle \left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)\)
  4. \(\displaystyle \left(\dfrac{\sqrt{3}}{2},-\dfrac{1}{2}\right)\)
  5. \(\displaystyle \left(\dfrac{-\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)\)
Answer.
A
(c)
How does the point you’ve just found compare with the point \((x,y)\text{,}\) where the terminal edge of \(\theta=\dfrac{4\pi}{3}\) intersects the unit circle?
  1. The \(x\)-values and the \(y\)-values are switched with each other.
  2. The \(x\)-values will be the same, but the \(y\)-values will have opposite signs.
  3. The \(y\)-values will be the same, but the \(x\)-values will have opposite signs.
  4. The \(x\)-values and the \(y\)-values will both have opposite signs.
Answer.
C
(d)
What are the sine and cosine of \(\theta=\dfrac{4\pi}{3}\text{?}\)
  1. \(\sin \dfrac{4\pi}{3} = \dfrac{1}{2}\) and \(\cos \dfrac{4\pi}{3} = \dfrac{\sqrt{3}}{2}\)
  2. \(\sin \dfrac{4\pi}{3} = -\dfrac{\sqrt{3}}{2}\) and \(\cos \dfrac{4\pi}{3} = -\dfrac{1}{2}\)
  3. \(\sin \dfrac{4\pi}{3} = -\dfrac{\sqrt{3}}{2}\) and \(\cos \dfrac{4\pi}{3} = \dfrac{1}{2}\)
  4. \(\sin \dfrac{4\pi}{3} = \dfrac{1}{2}\) and \(\cos \dfrac{4\pi}{3} = -\dfrac{\sqrt{3}}{2}\)
  5. \(\sin \dfrac{4\pi}{3} = -\dfrac{1}{2}\) and \(\cos \dfrac{4\pi}{3} = \dfrac{\sqrt{3}}{2}\)
Answer.
B

Activity 6.5.10.

Find \(\sin \theta\) and \(\cos \theta\) for each angle given.
(a)
\(\theta = \dfrac{\pi}{4}\)
Answer.
\(\sin \theta = \dfrac{\sqrt{2}}{2}\) and \(\cos \theta = \dfrac{\sqrt{2}}{2}\)
(b)
\(\theta = \dfrac{2\pi}{3}\)
Answer.
\(\sin \theta = \dfrac{\sqrt{3}}{2}\) and \(\cos \theta = -\dfrac{1}{2}\)
(c)
\(\theta = \dfrac{11\pi}{6}\)
Answer.
\(\sin \theta = -\dfrac{1}{2}\) and \(\cos \theta = \dfrac{\sqrt{3}}{2}\)
(d)
\(\theta = 135^\circ\)
Answer.
\(\sin \theta = \dfrac{sqrt{2}}{2}\) and \(\cos \theta = -\dfrac{\sqrt{2}}{2}\)

Activity 6.5.11.

Find the following for each angle graphed below.
  • \(\theta\) in radians and degrees
  • \(\displaystyle \sin \theta\)
  • \(\displaystyle \cos \theta\)
  • \(\displaystyle \tan \theta\)
  • \(\displaystyle \sec \theta\)
  • \(\displaystyle \csc \theta\)
  • \(\displaystyle \cot \theta\)
(a)
Answer.
  • \(\theta=\dfrac{5\pi}{6}\) or \(150^\circ\)
  • \(\displaystyle \sin \theta = \dfrac{1}{2}\)
  • \(\displaystyle \cos \theta = -\dfrac{\sqrt{3}}{2}\)
  • \(\tan \theta = -\dfrac{\sqrt{3}}{3}\) or \(-\dfrac{1}{\sqrt{3}}\)
  • \(\displaystyle \sec \theta = -\sqrt{3}\)
  • \(\displaystyle \csc \theta = -\dfrac{2\sqrt{3}}{3}\)
  • \(\displaystyle \cot \theta = 2\)
(b)
Answer.
  • \(\theta=\dfrac{5\pi}{4}\) or \(225^\circ\)
  • \(\displaystyle \sin \theta = -\dfrac{\sqrt{2}}{2}\)
  • \(\displaystyle \cos \theta = -\dfrac{\sqrt{2}}{2}\)
  • \(\tan \theta = 1\) or \(-\dfrac{1}{\sqrt{3}}\)
  • \(\displaystyle \sec \theta = -\sqrt{2}\)
  • \(\displaystyle \csc \theta = -\sqrt{2}\)
  • \(\displaystyle \cot \theta = 1\)

Remark 6.5.12.

So far we’ve only dealt with angles that are part of a special right triangle (30-60-90 or 45-45-90) or are a multiple of one of these angles, but we can extend to other angles as well.

Activity 6.5.13.

A point \((x,y)\) lies on the unit circle in Quadrant IV. Its \(x\)-coordinate is \(\dfrac{3}{4}\text{.}\)
(a)
Draw a sketch of the angle \(\theta\) whose terminal side intersects the unit circle as described above.
Answer.
(b)
What sign will the \(y\)-coordinate be?
  1. positive
  2. negative
Answer.
B
(c)
Find the exact value of the \(y\)-coordinate.
  1. \(\displaystyle \dfrac{7}{16}\)
  2. \(\displaystyle -\dfrac{7}{16}\)
  3. \(\displaystyle \dfrac{\sqrt{7}}{4}\)
  4. \(\displaystyle -\dfrac{\sqrt{7}}{4}\)
Answer.
D
(d)
Find \(\sin \theta\text{,}\) \(\cos \theta\text{,}\) \(\tan \theta\text{,}\) \(\sec \theta\text{,}\) \(\csc \theta\text{,}\) and \(\cot \theta\text{.}\)
Answer.
  • \(\displaystyle \sin \theta = -\dfrac{\sqrt{7}}{4}\)
  • \(\displaystyle \cos \theta = \dfrac{3}{4}\)
  • \(\displaystyle \tan \theta = -\dfrac{\sqrt{7}}{3}\)
  • \(\displaystyle \sec \theta = \dfrac{4}{3}\)
  • \(\csc \theta = -\dfrac{4}{\sqrt{7}}\) or \(-\dfrac{4\sqrt{7}}{7}\)
  • \(\cot \theta = -\dfrac{3}{\sqrt{7}}\) or \(-\dfrac{3\sqrt{7}}{7}\)

Activity 6.5.14.

Let \(\theta\) be the angle whose terminal side intersects the unit circle at the point described in each situation below. Find \(\sin \theta\text{,}\) \(\cos \theta\text{,}\) \(\tan \theta\text{,}\) \(\sec \theta\text{,}\) \(\csc \theta\text{,}\) and \(\cot \theta\text{.}\)
(a)
The point \(\left(\dfrac{4}{5},y\right)\) that lies on the unit circle in Quadrant I.
Answer.
  • \(\displaystyle \sin \theta = \dfrac{3}{5}\)
  • \(\displaystyle \cos \theta = \dfrac{4}{5}\)
  • \(\displaystyle \tan \theta = \dfrac{3}{4}\)
  • \(\displaystyle \sec \theta = \dfrac{5}{4}\)
  • \(\displaystyle \csc \theta = \dfrac{5}{3}\)
  • \(\displaystyle \cot \theta = \dfrac{4}{3}\)
(b)
The point \(\left(-\dfrac{35}{37},\dfrac{12}{37}\right)\text{.}\)
Answer.
  • \(\displaystyle \sin \theta = \dfrac{12}{37}\)
  • \(\displaystyle \cos \theta = -\dfrac{35}{37}\)
  • \(\displaystyle \tan \theta = -\dfrac{12}{35}\)
  • \(\displaystyle \sec \theta = -\dfrac{37}{35}\)
  • \(\displaystyle \csc \theta = \dfrac{37}{12}\)
  • \(\displaystyle \cot \theta = -\dfrac{35}{12}\)
(c)
The point \(\left(x,-\dfrac{\sqrt{11}}{6}\right)\) that lies on the unit circle in Quadrant III.
Answer.
  • \(\displaystyle \sin \theta = -\dfrac{\sqrt{11}}{6}\)
  • \(\displaystyle \cos \theta = -\dfrac{5}{6}\)
  • \(\displaystyle \tan \theta = \dfrac{\sqrt{11}}{5}\)
  • \(\displaystyle \sec \theta = -\dfrac{6}{5}\)
  • \(\csc \theta = -\dfrac{6}{\sqrt{11}}\) or \(-\dfrac{6\sqrt{11}}{11}\)
  • \(\cot \theta = \dfrac{5}{\sqrt{11}}\) or \(\dfrac{5\sqrt{11}}{11}\)
(d)
The point \(\left(\dfrac{12}{13},-\dfrac{5}{13}\right)\text{.}\)
Answer.
  • \(\displaystyle \sin \theta = -\dfrac{5}{13}\)
  • \(\displaystyle \cos \theta = \dfrac{12}{13}\)
  • \(\displaystyle \tan \theta = -\dfrac{5}{12}\)
  • \(\displaystyle \sec \theta = \dfrac{13}{12}\)
  • \(\displaystyle \csc \theta = -\dfrac{13}{5}\)
  • \(\displaystyle \cot \theta = -\dfrac{12}{5}\)

Exercises 6.5.2 Exercises

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