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Precalculus for Team-Based Inquiry Learning: PREVIEW Edition

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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 is the amount of a circle between the two rays.

Activity 6.1.2.

We know that if you complete a full turn of the circle the angle created will be 360 degrees. Use this to estimate the measure of the given angles.
(a)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
(b)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)
(c)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)

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.

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

Remark 6.1.5.

Activity or remark - Something about the circumference of a circle being another way to measure the angle. \(C=2\pi r\) divide both sides by the radius, so a full circle or \(360^{\circ}=2\pi\) radians

Definition 6.1.6.

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

Activity 6.1.7.

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. \(\displaystyle \frac{\pi}{2}\)
(b)
\(45^{\circ}\)
  1. \(\displaystyle \frac{\pi}{4}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{4}\)
  4. \(\displaystyle \frac{\pi}{2}\)

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 degrees.
(a)
\(\frac{\pi}{2}\)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 180^{\circ}\)
  4. \(\displaystyle 360^{\circ}\)
(b)
\(\frac{3\pi}{4}\)
  1. \(\displaystyle 45^{\circ}\)
  2. \(\displaystyle 90^{\circ}\)
  3. \(\displaystyle 135^{\circ}\)
  4. \(\displaystyle 180^{\circ}\)

Subsection 6.1.2 Exercises

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