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

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Section 6.1 Degree and Radian Measure (TR1)

Subsection 6.1.1 Activities

Definition 6.1.1.

An angle is formed by joining two rays at their starting points. The point where they are joined is called the vertex of the angle. The measure of an angle describes the amount of rotation between the two rays.

Activity 6.1.2.

An angle that is rotated all the way around back to its starting point measures \(360^\circ\text{,}\) like a circle. Use this to estimate the measure of the given angles.
(a)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
Answer.
B
(b)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
Answer.
D
(c)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
Answer.
C

Definition 6.1.3.

An angle is in standard position if its vertex is located at the origin and its initial side extends along the positive \(x\)-axis.
An angle measured counterclockwise from the initial side has a positive measure, while an angle measured clockwise from the initial side has a negative measure.

Activity 6.1.4.

Estimate the measure of the angles drawn in standard position.
(a)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
Answer.
A
(b)
  1. \(\displaystyle 180^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle -180^{\circ}\)
  4. \(\displaystyle -90^{\circ}\)
Answer.
C
(c)
  1. \(\displaystyle 30^{\circ}\)
  2. \(\displaystyle -150^{\circ}\)
  3. \(\displaystyle -210^{\circ}\)
  4. \(\displaystyle 210^{\circ}\)
Answer.
B
(d)
Draw an angle of measure \(-225^{\circ} \) in standard position.
Answer.

Remark 6.1.5.

Degrees are not the only way to measure an angle. We can also describe the angle’s measure by the amount of the circumference of the circle that the angle’s rotation created. We’ll need to define a few terms to help us come up with this new measurement.

Definition 6.1.6.

A central angle is an angle whose vertex is at the center of a circle.

Definition 6.1.7.

One radian is the measure of a central angle of a circle that intersects an arc the same length as the radius.

Observation 6.1.8.

Recall that the circumference of a circle is given by \(C=2\pi r\text{,}\) where \(r\) is the radius of the circle. That means if we rotate through an entire circle, the circumference is \(2\pi r\) which implies that the angle was \(2\pi\) radians. Thus \(2\pi\) radians is the same measure as \(360^\circ\text{.}\)

Activity 6.1.9.

We now know that one turn around the circle measures \(360^{\circ}\) and also \(2\pi\) radians. Use this information to set up a proportion to find the equivalent radian measure of the following angles that are given in degrees.
(a)
\(180^{\circ}\)
Hint.
Try setting up a proportion! \(\dfrac{180^\circ}{360^\circ}= \dfrac{x}{2\pi}\)
  1. \(\displaystyle \frac{\pi}{4}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{4}\)
  4. \(\displaystyle \frac{\pi}{2}\)
Answer.
B
(b)
\(45^{\circ}\)
  1. \(\displaystyle \frac{\pi}{4}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{4}\)
  4. \(\displaystyle \frac{\pi}{2}\)
Answer.
A

Activity 6.1.10.

Continue using the fact that one turn around the circle measures \(360^{\circ}\) and also \(2\pi\) radians. Use this information to set up a proportion to find the equivalent degree measure of the following angles that are given in radians.
(a)
\(\dfrac{\pi}{2}\)
Hint.
Try setting up a proportion! \(\dfrac{x}{360^\circ}= \dfrac{\frac{\pi}{2}}{2\pi}\)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 180^{\circ}\)
  4. \(\displaystyle 360^{\circ}\)
Answer.
B
(b)
\(\frac{3\pi}{4}\)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
Answer.
C

Activity 6.1.11.

We’ll now use the proportions from before to come up with a way to convert between degrees and radians for any given angle. We’ll call \(a\) the angle’s measure in degrees and \(b\) the angle’s measure in radians. So, we have the following proportion that must hold:
\begin{equation*} \dfrac{a}{360^\circ}= \dfrac{b}{2\pi} \end{equation*}

(a)

Let’s say we know an angle’s measure in degrees, \(a\text{,}\) and need to find the angle’s measure in radians, \(b\text{.}\) Solve for \(b\) in the proportion.
Answer.
\(b=\dfrac{\pi}{180^\circ}\cdot a\)

(b)

Use the formula you just developed to convert \(60^\circ\) to radians. Leave your answer in terms of \(\pi\text{.}\) Do not approximate!
Answer.
\(\dfrac{\pi}{3}\)

(c)

Now let’s assume we know an angle’s measure in radians, \(b\text{,}\) and need to find the angle’s measure in degrees, \(a\text{.}\) Solve for \(a\) in the proportion.
Answer.
\(a=\dfrac{180^\circ}{\pi}\cdot b\)

(d)

Use the formula you just developed to convert \(\dfrac{\pi}{6}\) to degrees.
Answer.
\(30^\circ\)

Remark 6.1.12.

We now have a way to convert back and forth between the two types of measurements. If we know the angle’s measure in degrees, we multiply it by \(\dfrac{\pi}{180^\circ}\) to find the measure in radians. If we know the angle’s measure in radians, we multiply it by \(\dfrac{180^\circ}{\pi}\) to find the measure in degrees.

Activity 6.1.13.

Convert each of the following angles.

(a)

\(\dfrac{2\pi}{3}\) radians to degrees
Answer.
\(120^\circ\)

(b)

\(\dfrac{7\pi}{6}\) radians to degrees
Answer.
\(210^\circ\)

(c)

\(240^\circ \) to radians
Answer.
\(\dfrac{4\pi}{3}\)

(d)

\(315^\circ \) to radians
Answer.
\(\dfrac{7\pi}{4}\)

Subsection 6.1.2 Exercises

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