Radian measure

​​In a nutshell

So far you have measured angles in the units of degrees. Radians are an alternative unit for measuring angles, and give a much more natural description of an angle (whereas the idea of a full rotation being $$360 \degree$$ is arbitrary). It is important to know the definition of a radian, how to convert between degrees and radians and also how to use angles measured in radians in trigonometric problems.

Radians

Definition

An angle $$\theta$$, in radians, is defined as the arc length, subtended by the angle $$\theta$$, divided by the radius of a circle.

​$$\boxed{\theta = \dfrac {l}{r}}$$​​

​$$\theta$$ is the angle in radians.​ ​$$l$$ is the arc length.​ ​$$r$$ is the radius.​

Consider a circle where the arc length is the circumference, so that the angle subtending the circumference gives one full rotation.

Since the arc length for one full rotation is the circumference, substitute in $$C=2\pi r$$ to give: $$\theta = \dfrac {l}{r} = \dfrac {2 \pi r}{r} = 2 \pi$$ This means that one full rotation is an angle of $$2 \pi$$ in radians. This gives the result: $$\boxed{360 \degree = 2 \pi \ radians}$$ or $$\boxed{180 \degree = \pi \ radians}$$​​

Converting between degrees and radians

The conversion $$180 \degree = \pi \ radians$$ can now be used to convert between degrees and radians.

$$\begin{aligned}& \xrightarrow{ \times \dfrac{\pi}{180}} \\\Large 180 \degree & & \Large {\pi \ radians } \\& \xleftarrow{ \times \dfrac{180}{\pi}} \\\end {aligned}$$​​

To convert degrees to radians, $$\times \dfrac{\pi}{180}$$ and to convert radians to degrees, $$\times \dfrac{180}{\pi}$$.

Example 1

What is $$90\degree$$ in radians?

Convert from degrees into radians:

​$$90 \degree = 90 \times \dfrac {\pi}{180} =\underline{ \dfrac {\pi}{2} \ radians}$$​​

Note: This is a common value that should be memorised.

Example 2

What is $$1$$ radian in degrees?​

Convert from radians to degrees:

​$$1 \ radian = 1 \times \dfrac{180}{\pi} = \underline{57.3 \degree \ (3 \ s.f.)}$$​​

Trigonometric graphs

You should be able to sketch all the trig functions with the angle in radians on the $$x$$-axis. Note the main angles in radians: $$90 \degree = \frac {\pi} {2} \ rads$$, $$180 \degree = \pi \ rads$$, $$270 \degree = \frac {3\pi} {2} \ rads$$ and $$360 \degree = 2 \pi \ rads$$.

​$$y=\sin(x)$$​​	​$$y= \cos(x)$$​​	​$$y=\tan (x)$$​​