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

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Section 6.2 Angle Position and Arc Length (TR2)

Subsection 6.2.1 Activities

Activity 6.2.1.

Consider the angle given below:
Which of the following angles describe the plotted angle?
  1. \(\displaystyle -45^\circ\)
  2. \(\displaystyle -135^\circ\)
  3. \(\displaystyle -225^\circ\)
  4. \(\displaystyle -315^\circ\)
Answer.
D.

Definition 6.2.2.

Two angles are called coterminal angles if they have the same terminal side when drawn in standard position.

Activity 6.2.3.

Consider the angle given below:
(a)
Find two angles larger than \(60^\circ\) that are coterminal to \(60^\circ\text{.}\)
Answer.
\(420^\circ\text{,}\) \(780^\circ\text{,}\) among others.
(b)
Find two angles smaller than \(60^\circ\) that are coterminal to \(60^\circ\text{.}\)
Answer.
\(-300^\circ\text{,}\) \(-660^\circ\text{,}\) among others.

Observation 6.2.4.

For any angle \(\theta\text{,}\) the angle \(\theta + k\cdot 360^\circ\) is coterminal to \(\theta\) for any integer \(k\text{.}\)

Remark 6.2.5.

Since there are many coterminal angles for any given angle, it is convenient to systematically choose one for every angle. For a given angle, we typically choose the smallest positive coterminal angle to work with instead.

Definition 6.2.6.

If \(\theta\) is an angle, there is a unique angle \(\alpha\) with \(0 \leq \alpha \lt 360^\circ\) (or \(0\leq \alpha \lt 2\pi\)) such that \(\alpha\) and \(\theta\) are coterminal. This angle \(\alpha\) is called the principal angle of \(\theta\text{.}\)

Activity 6.2.7.

Find the principal angles for each of the following angles.
(a)
\(540^\circ\)
Answer.
\(180^\circ\)
(b)
\(-600^\circ\)
Answer.
\(120^\circ\)
(c)
\(\dfrac{11\pi}{3}\)
Answer.
\(\dfrac{5\pi}{3}\)
(d)
\(\dfrac{-7\pi}{5}\)
Answer.
\(\dfrac{3\pi}{5}\)

Remark 6.2.8.

Recall that the circumference of a circle of radius \(r\) is \(2 \pi r\text{.}\) We will use this to determine the lengths of arcs on a circle.

Activity 6.2.9.

Consider the portion of a circle of radius \(1\) graphed below.
(a)
What is the circumference of an entire circle of radius \(1\text{?}\)
  1. \(\displaystyle \frac{\pi}{2}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{2}\)
  4. \(\displaystyle 2\pi\)
Answer.
D.
(b)
What portion of the entire circle is the sector graphed above?
  1. \(\displaystyle \frac{1}{4}\)
  2. \(\displaystyle \frac{1}{3}\)
  3. \(\displaystyle \frac{1}{2}\)
  4. \(\displaystyle \frac{3}{4}\)
Answer.
A.
(c)
Use proportions to determine the length of the arc displayed above.
  1. \(\displaystyle \frac{\pi}{2}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{2}\)
  4. \(\displaystyle 2\pi\)
Answer.
A

Activity 6.2.10.

Consider the portion of a circle of radius \(3\) graphed below.
(a)
What is the circumference of an entire circle of radius \(3\text{?}\)
  1. \(\displaystyle \pi\)
  2. \(\displaystyle 3\pi\)
  3. \(\displaystyle 6\pi\)
  4. \(\displaystyle 9\pi\)
Answer.
C.
(b)
What portion of the entire circle is the sector graphed above?
  1. \(\displaystyle \frac{1}{12}\)
  2. \(\displaystyle \frac{1}{6}\)
  3. \(\displaystyle \frac{1}{4}\)
  4. \(\displaystyle \frac{1}{3}\)
Answer.
B.
(c)
Use proportions to determine the length of the arc displayed above.
  1. \(\displaystyle \frac{\pi}{2}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle 2\pi\)
  4. \(\displaystyle 3\pi\)
Answer.
B.

Observation 6.2.11.

For a sector of angle \(\theta\) and radius \(r\text{,}\) we can use proportions to find the length of the arc, \(s\text{.}\) If \(\theta\) is measured in degrees, we have \(s=\frac{\theta}{360^\circ}\left(2\pi r\right)\text{,}\) which simplifies to
\begin{equation*} s=\frac{\theta}{180^\circ}\pi r\text{.} \end{equation*}
In radians, the formula is even nicer: \(s=\frac{\theta}{2\pi}\left(2 \pi r\right)\text{,}\) which simplifies to
\begin{equation*} s=\theta r\text{.} \end{equation*}

Activity 6.2.12.

Find the lengths of the arcs described below.
(a)
The length of the arc of a sector of measure \(120^\circ\) of a circle of radius \(4\text{.}\)
Answer.
\(\frac{8\pi}{3}\)
(b)
The length of the arc of a sector of measure \(270^\circ\) of a circle of radius \(2\text{.}\)
Answer.
\(3\pi\)
(c)
The length of the arc of a sector of measure \(\dfrac{5\pi}{6}\) of a circle of radius \(3\text{.}\)
Answer.
\(\frac{5\pi}{2}\)
(d)
The length of the arc of a sector of measure \(\dfrac{11\pi}{12}\) of a circle of radius \(6\text{.}\)
Answer.
\(\frac{11\pi}{2}\)

Remark 6.2.13.

Recalling that the area of a circle of radius \(r\) is \(\pi r^2\text{,}\) we can use this same idea of proportions to find the area of a sector of a circle.

Activity 6.2.14.

Consider the portion of a circle of radius \(1\) graphed below.
(a)
What is the area of an entire circle of radius \(1\text{?}\)
  1. \(\displaystyle \frac{\pi}{2}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{2}\)
  4. \(\displaystyle 2\pi\)
Answer.
B.
(b)
What portion of the entire circle is the sector graphed above?
  1. \(\displaystyle \frac{1}{4}\)
  2. \(\displaystyle \frac{1}{3}\)
  3. \(\displaystyle \frac{1}{2}\)
  4. \(\displaystyle \frac{3}{4}\)
Answer.
A.
(c)
Use proportions to determine the area of the arc displayed above.
  1. \(\displaystyle \frac{\pi}{4}\)
  2. \(\displaystyle \frac{\pi}{2}\)
  3. \(\displaystyle \pi\)
  4. \(\displaystyle \frac{3\pi}{2}\)
Answer.
A.

Activity 6.2.15.

Consider the portion of a circle of radius \(3\) graphed below.
(a)
What is the area of an entire circle of radius \(3\text{?}\)
  1. \(\displaystyle \pi\)
  2. \(\displaystyle 3\pi\)
  3. \(\displaystyle 6\pi\)
  4. \(\displaystyle 9\pi\)
Answer.
D.
(b)
What portion of the entire circle is the sector graphed above?
  1. \(\displaystyle \frac{1}{12}\)
  2. \(\displaystyle \frac{1}{6}\)
  3. \(\displaystyle \frac{1}{4}\)
  4. \(\displaystyle \frac{1}{3}\)
Answer.
B.
(c)
Use proportions to determine the area of the sector displayed above.
  1. \(\displaystyle \frac{\pi}{2}\)
  2. \(\displaystyle \pi\)
  3. \(\displaystyle \frac{3\pi}{2}\)
  4. \(\displaystyle 2\pi\)
Answer.
C.

Observation 6.2.16.

For a sector of angle \(\theta\) and radius \(r\text{,}\) we can use proportions to find the area of the arc. If \(\theta\) is measured in degrees, we have \(A=\frac{\theta}{360^\circ}\left(\pi r^2\right)\text{.}\) In radians, the formula is even nicer: \(A=\frac{\theta}{2\pi}\left(\pi r^2\right)\text{,}\) which simplifies to
\begin{equation*} A=\frac{1}{2}\theta r^2\text{.} \end{equation*}

Activity 6.2.17.

Find the areas of each sector described below.
(a)
The sector with central angle \(120^\circ\) in a circle of radius \(4\text{.}\)
Answer.
\(\dfrac{16\pi}{3}\)
(b)
The sector with central angle \(270^\circ\) in a circle of radius \(2\text{.}\)
Answer.
\(3\pi\)
(c)
The sector with central angle \(\dfrac{5\pi}{6}\) in a circle of radius \(3\text{.}\)
Answer.
\(\dfrac{15\pi}{4}\)
(d)
The sector with central angle \(\dfrac{11\pi}{12}\) in a circle of radius \(6\text{.}\)
Answer.
\(\dfrac{33\pi}{2}\)

Exercises 6.2.2 Exercises

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