Degree and Radian Angle Measure
Instructions
- The first videos below explain the concepts in this section.
- This page includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
When you are done, you can use the graph to pick another section or use the buttons to go to the next section
Learning Objectives
- Defining radians
- How to convert between radians and degrees
Concept Video(s)
Exercises
1.
Answer: \(135^\circ\) is equivalent to \(\dfrac{3\pi}{4}\) radians.
\(300^\circ\) is equivalent to \(\dfrac{5\pi}{3}\) radians.
Solution Method: To convert from degrees to radians, we use the formula\begin{equation*}
\text{degrees } \times \frac{\pi}{180} = \text{ radians}
\end{equation*}This conversion factor comes from the fact that\begin{equation*}
2\pi \text{ radians} = 360^\circ
\end{equation*}Then we have\begin{align*}
\pi \text{ radians} &= 180^\circ\\
\frac{\pi}{180} &= 1^\circ
\end{align*}Now, to convert \(135^\circ,\)\begin{align*}
135^\circ &= 135 \times \frac{\pi}{180} \text{ radians}\\
&= \frac{135\pi}{180} \text{ radians}\\
&= \frac{27\pi}{36} \text{ radians}\\
&= \frac{3\pi}{4} \text{ radians}
\end{align*}And\begin{align*}
300^\circ &= 300 \times \frac{\pi}{180} \text{ radians}\\
&= \frac{300\pi}{180} \text{ radians}\\
&= \frac{10\pi}{6} \text{ radians}\\
&= \frac{5\pi}{3} \text{ radians}
\end{align*}
\(300^\circ\) is equivalent to \(\dfrac{5\pi}{3}\) radians.
Solution Method: To convert from degrees to radians, we use the formula\begin{equation*}
\text{degrees } \times \frac{\pi}{180} = \text{ radians}
\end{equation*}This conversion factor comes from the fact that\begin{equation*}
2\pi \text{ radians} = 360^\circ
\end{equation*}Then we have\begin{align*}
\pi \text{ radians} &= 180^\circ\\
\frac{\pi}{180} &= 1^\circ
\end{align*}Now, to convert \(135^\circ,\)\begin{align*}
135^\circ &= 135 \times \frac{\pi}{180} \text{ radians}\\
&= \frac{135\pi}{180} \text{ radians}\\
&= \frac{27\pi}{36} \text{ radians}\\
&= \frac{3\pi}{4} \text{ radians}
\end{align*}And\begin{align*}
300^\circ &= 300 \times \frac{\pi}{180} \text{ radians}\\
&= \frac{300\pi}{180} \text{ radians}\\
&= \frac{10\pi}{6} \text{ radians}\\
&= \frac{5\pi}{3} \text{ radians}
\end{align*}
2.
Answer: \(\dfrac{\pi}{3}\) radians is equivalent to \(60^\circ\)
\(\dfrac{7\pi}{4}\) radians is equivalent to \(315^\circ\)
Solution Method: To convert from radians to degrees we use the formula \begin{equation*}
\text{radians } \times \frac{180}{\pi} = \text{ degrees}
\end{equation*}This conversion factor comes from the fact that \begin{equation*}
2\pi \text{ radians} = 360^\circ
\end{equation*}Then we have\begin{align*}
\pi \text{ radians} &= 180^\circ\\
1 \text{ radian} &= \frac{180^\circ}{\pi}
\end{align*}So we have \begin{align*}
\frac{\pi}{3} \text{ radians} &= \frac{\pi}{3} \times \frac{180}{\pi} \text{ degrees}\\
&= \frac{180}{3} \text{ degrees} \\
&= 60 ^\circ
\end{align*}And\begin{align*}
\frac{7\pi}{4} \text{ radians} &= \frac{7\pi}{4} \times \frac{180}{\pi} \text{ degrees}\\
&= \frac{7(180)}{4} \text{ degrees} \\
&= 7(45)^\circ \\
&= 315^\circ
\end{align*}
\(\dfrac{7\pi}{4}\) radians is equivalent to \(315^\circ\)
Solution Method: To convert from radians to degrees we use the formula \begin{equation*}
\text{radians } \times \frac{180}{\pi} = \text{ degrees}
\end{equation*}This conversion factor comes from the fact that \begin{equation*}
2\pi \text{ radians} = 360^\circ
\end{equation*}Then we have\begin{align*}
\pi \text{ radians} &= 180^\circ\\
1 \text{ radian} &= \frac{180^\circ}{\pi}
\end{align*}So we have \begin{align*}
\frac{\pi}{3} \text{ radians} &= \frac{\pi}{3} \times \frac{180}{\pi} \text{ degrees}\\
&= \frac{180}{3} \text{ degrees} \\
&= 60 ^\circ
\end{align*}And\begin{align*}
\frac{7\pi}{4} \text{ radians} &= \frac{7\pi}{4} \times \frac{180}{\pi} \text{ degrees}\\
&= \frac{7(180)}{4} \text{ degrees} \\
&= 7(45)^\circ \\
&= 315^\circ
\end{align*}