Chapter 3, Problem 1RE

What is the meaning of “an angle with measure 2 radians"?

To determine

The meaning of the statement “an angle with measure 2 radians.”

Answer to Problem 1RE

Solution: An angle measuring 2 radians is approximately equal to an angle that measures ${114.6}^{\circ }$ and basically represents a central angle of a circle that intercepts an arc of length equal to twice the radius of the circle.

Explanation of Solution

An angle with measure 2 radians intercepts an arc of length that is 2 times of the radius of the circle.

Thus, if l is the length of the arc and r is the radius of the circle, we can write

$\begin{array}{c}l=r\theta \\ l=r\cdot 2\\ =2r\end{array}$

Also, it is known that

$1°=\frac{\pi }{180}\text{ radian}$

and

$\begin{array}{c}1\text{ radian}=\left(\frac{180}{\pi }\right)\text{ degree}\\ \text{2 radians}=2\left(\frac{180}{\pi }\right)\text{ degree}\\ ={114.6}^{\circ }\end{array}$

An angle measuring 2 radians is equivalent to an angle measuring ${114.6}^{\circ }$.