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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.

Activity 6.1.8.

Using the fact that one turn around the circle is \(360^{\circ}\) and also \(2\pi\) radians. Find the measure of the following angles in radians.
(a)
\(180^{\circ}\)
  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 6.1.9.

Using the fact that one turn around the circle is \(360^{\circ}\) and also \(2\pi\) radians. Find the measure of the following angles in degrees.
(a)
\(\frac{\pi}{2}\)
  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

Subsection 6.1.2 Exercises

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