Radians are a useful unit for measuring angles in trigonometry. In the same way as you can express temperature in Celsius or in Fahrenheit, an angle can be expressed in degrees or radians.

Theory

$$\text{Angleinradians}=\frac{\text{arclength}}{\text{radius}}=\frac{b}{r}$$ |

You know that there are $360$° in a circle. Likewise, there are $2\pi $ radians in a circle. Thus, $360\text{\xb0}=2\pi $, which means that $\pi =180\text{\xb0}$. To switch between degrees and radians, you can use the following formulas:

Formula

$$\begin{array}{cc}{n}^{\circ}=\frac{180\text{\xb0}}{\pi}\cdot {\nu}_{\text{rad}}& \\ {\nu}_{\text{rad}}=\frac{\pi}{180\text{\xb0}}\cdot {n}^{\circ}& \end{array}$$

$${n}^{\circ}=\frac{180\text{\xb0}}{\pi}\cdot {\nu}_{\text{rad}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\nu}_{\text{rad}}=\frac{\pi}{180\text{\xb0}}\cdot {n}^{\circ}$$ |

You can see from the two formulas that they are simply two versions of the same expression. To transform one into the other, you solve the equation with respect to the other angle measurement.

**Note!** Remember to change your calculator settings from degrees to radians (or vice versa) each time you swap between them.

Example 1

**You are given an angle of $\text{}37\text{}\text{\xb0}$. What is this angle equal to in radians? **

Putting this into the formula gives:

$${\nu}_{\text{rad}}=\frac{\pi}{180\text{\xb0}}\cdot 37\text{\xb0}\approx 0.646$$ |

Example 2

**You are given an angle of $\frac{2\pi}{3}$, what is this in degrees? **

The formula gives you:

$${n}^{\circ}=\frac{180\text{\xb0}}{\pi}\cdot \frac{2\pi}{3}=120\text{\xb0}$$ |