6.1 Angle Measure
6.1.1 Learning Objectives
By the end of this section, you should be able to:
- convert between degrees and radians;
- identify coterminal angles;
- find arc length and sector area;
- calculate angular and linear speed; and
- interpret angle measures in context.
6.1.2 Degrees and Radians
One complete revolution is \(360^\circ=2\pi\) radians. Therefore,
\[ 180^\circ=\pi\text{ radians}. \]
Multiply degrees by \(\pi/180\) to obtain radians, and multiply radians by \(180/\pi\) to obtain degrees.
Example 6.1 Converting Angle Measures
Convert \(135^\circ\) to radians and \(5\pi/6\) to degrees.
Solution
\[ 135^\circ\left(\frac{\pi}{180^\circ}\right)=\frac{3\pi}{4}, \qquad \frac{5\pi}{6}\left(\frac{180^\circ}{\pi}\right)=150^\circ. \]
\[ \boxed{135^\circ=3\pi/4;\quad 5\pi/6=150^\circ} \]
Exercise 6.1 Convert \(300^\circ\) to radians.
Show answer
Answer: \(\boxed{5\pi/3}\)
6.1.3 Coterminal Angles
Angles with the same initial and terminal sides are coterminal. In degrees they differ by \(360^\circ k\); in radians they differ by \(2\pi k\), where \(k\in\mathbb Z\).
Example 6.2 Finding Coterminal Angles
Find a positive and a negative angle coterminal with \(70^\circ\).
Solution
\[ 70^\circ+360^\circ=430^\circ,\qquad 70^\circ-360^\circ=-290^\circ. \]
\[ \boxed{430^\circ\text{ and }-290^\circ} \]
Exercise 6.2 Find a coterminal angle in \([0,2\pi)\) for \(17\pi/6\).
Show answer
Answer: \(\boxed{5\pi/6}\)
6.1.4 Arc Length and Sector Area
For a circle of radius \(r\) and central angle \(\theta\) measured in radians,
\[ s=r\theta \]
is the arc length, and
\[ A=\frac12r^2\theta \]
is the sector area.
Example 6.3 Finding Arc Length and Area
A circle has radius \(8\text{ cm}\) and central angle \(3\pi/5\). Find the arc length and sector area.
Solution
\[ s=8\left(\frac{3\pi}{5}\right)=\frac{24\pi}{5}\text{ cm}, \]
\[ A=\frac12(8)^2\left(\frac{3\pi}{5}\right)=\frac{96\pi}{5}\text{ cm}^2. \]
\[ \boxed{s=24\pi/5\text{ cm};\quad A=96\pi/5\text{ cm}^2} \]
Exercise 6.3 Find the arc length when \(r=12\text{ m}\) and \(\theta=5\pi/6\).
Show answer
Answer: \(\boxed{10\pi\text{ m}}\)
6.1.5 Angular and Linear Speed
If an object rotates through angle \(\theta\) in time \(t\), its angular speed is
\[ \omega=\frac{\theta}{t}. \]
At radius \(r\), its linear speed is
\[ v=r\omega. \]
Example 6.4 Relating Angular and Linear Speed
A wheel of radius \(0.4\text{ m}\) rotates at \(6\pi\text{ rad/s}\). Find the rim speed.
Solution
\[ v=r\omega=0.4(6\pi)=2.4\pi. \]
\[ \boxed{v=2.4\pi\text{ m/s}\approx7.54\text{ m/s}} \]
Exercise 6.4 A point \(0.25\text{ m}\) from the centre rotates at \(8\text{ rad/s}\). Find its linear speed.
Show answer
Answer: \(\boxed{2\text{ m/s}}\)
6.1.6 Conceptual Takeaways
- Degrees and radians describe the same rotation using different units.
- Radian measure connects angles directly to arc length.
- Coterminal angles differ by complete revolutions.
- Sector formulas require radians.
- Linear speed depends on both angular speed and radius.
6.1.7 Skills You Should Be Able to Do
- Convert degrees and radians.
- Find coterminal angles.
- Calculate arc length and sector area.
- Determine angular and linear speeds.
- Include correct units in circular-motion problems.
6.1.8 Practice Problems with Solutions
Convert \(225^\circ\) to radians.
Show Solution
\[ 225^\circ\left(\frac{\pi}{180^\circ}\right)=\boxed{\frac{5\pi}{4}}. \]
Convert \(7\pi/12\) to degrees.
Show Solution
\[ \frac{7\pi}{12}\left(\frac{180^\circ}{\pi}\right)=\boxed{105^\circ}. \]
Find an angle in \([0,360^\circ)\) coterminal with \(-765^\circ\).
Show Solution
Add \(720^\circ\), then \(360^\circ\):
\[ -765^\circ+1080^\circ=\boxed{315^\circ}. \]
Find the arc length for \(r=9\text{ cm}\) and \(\theta=2\pi/3\).
Show Solution
\[ s=9(2\pi/3)=\boxed{6\pi\text{ cm}}. \]
Find the area of a sector with radius \(10\text{ m}\) and angle \(1.2\) radians.
Show Solution
\[ A=\frac12(10)^2(1.2)=\boxed{60\text{ m}^2}. \]
A wheel of radius \(0.35\text{ m}\) turns at \(10\text{ rad/s}\). Find the rim speed.
Show Solution
\[ v=r\omega=0.35(10)=\boxed{3.5\text{ m/s}}. \]