2.2 Unit Circle
Convert each of the following quantities to the alternative measure: degrees to radians or radians to degrees.
a. [latex]30^{\circ}[/latex]
b. [latex]\frac{2\pi}{3}[/latex] radians
c. [latex]\frac{5\pi}{4}[/latex] radians
d. [latex]240^{\circ}[/latex]
e. [latex]17^{\circ}[/latex]
f. [latex]2[/latex] radians
Show Solution
a. [latex]30^{\circ}\cdot \frac{\pi}{180}=\frac{\pi}{6}[/latex]
b. [latex]\frac{2\pi}{3}\cdot \frac{180}{\pi}=120^{\circ}[/latex]
c. [latex]\frac{5\pi}{4}\cdot \frac{180}{\pi}=225^{\circ}[/latex]
d. [latex]240^{\circ}\cdot \frac{\pi}{180}=\frac{4\pi}{3}[/latex]
e. [latex]17^{\circ}\cdot \frac{\pi}{180}=\frac{17\pi}{180}[/latex]
f. [latex]2\cdot \frac{180}{\pi}=\frac{360}{\pi}^{\circ}[/latex]
Determine each of the following values or points exactly.
a. In a circle of radius 11, the arc length intercepted by a central angle of [latex]\frac{5\pi}{3}[/latex].
b. In a circle of radius 3, the central angle measure that intercepts an arc of length [latex]\frac{\pi}{4}[/latex].
c. The radius of the circle in which an angle of [latex]\frac{7\pi}{6}[/latex] intercepts an arc of length [latex]\frac{\pi}{2}[/latex].
d. The exact coordinates of the point on the circle of radius 5 that lies [latex]\frac{25\pi}{6}[/latex] units counterclockwise along the circle from [latex](5,0)[/latex].
Show Solution
Use formula [latex]s=r\cdot \theta[/latex].
a. [latex]r=11[/latex], [latex]\theta=\frac{5\pi}{3}[/latex]
[latex]s=11\cdot \frac{5\pi}{3}= \frac{55\pi}{3}[/latex]
b. [latex]r=3[/latex], [latex]s=\frac{\pi}{4}[/latex]
[latex]\frac{\pi}{4}=3 \cdot \theta[/latex]
[latex]\theta=\frac{\pi}{12}[/latex]
c. [latex]\theta=\frac{7\pi}{6}[/latex], [latex]s=\frac{\pi}{2}[/latex]
[latex]\frac{\pi}{2}=r\cdot \frac{7\pi}{6}[/latex]
[latex]r=\frac{3}{7}[/latex]
d. [latex]r=5[/latex], [latex]s=\frac{25\pi}{6}[/latex]
[latex]\frac{25\pi}{6}=5\cdot \theta[/latex]
[latex]\theta=\frac{5\pi}{6}[/latex]
Then the coordinate for a circle of radius 5 at this angle is [latex](-\frac{5\sqrt{3}}{2}, \frac{5}{2})[/latex].