How Do Radians Work?

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

Angle in Radians

An angle in radians

Angle in radians = \frac{arc length}{radius} = \frac{b}{r}

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

Formula

Converting Between Degrees and Radians

\begin{matrix} n^{\circ} = \frac{180^{\circ}}{π} \cdot ν_{rad} \\ ν_{rad} = \frac{π}{180^{\circ}} \cdot n^{\circ} \end{matrix}

n^{\circ} = \frac{180^{\circ}}{π} \cdot ν_{rad} and ν_{rad} = \frac{π}{180^{\circ}} \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 $37^{\circ}$ . What is this angle equal to in radians?

Putting this into the formula gives:

ν_{rad} = \frac{π}{180^{\circ}} \cdot 37^{\circ} \approx 0.646

Example 2

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

The formula gives you:

n^{\circ} = \frac{180^{\circ}}{π} \cdot \frac{2 π}{3} = 120^{\circ}

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How Does the Unit Circle Work?

What Is the Definition of Supplementary Angles?

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