Converting Between Radians and Degrees

Radians

A radian is another way to measure an angle.

Now, notice that the amount if radii that can fit on the circumference doesn't depend on the length of the radius, it is just always $2\pi$!

This is the basis of how Radians are defined.

Lets say that there is an angle $\theta$. If we have a circle with an arbitrary radius $r$ and we have a arc that corresponds to the angle. The number of radii that can fit on the arc is the radian measure of the angle.

Converting Degrees into Radians and back

For an angle of $x$ radians, you can denote that by $x\text{ rad}$, $x^{rad}$, or $x^c$

Now, using the definition above, we know that $360^\circ=2\pi \text{ rad}$.

So we have: $1^\circ=\frac{\pi}{180} \text{ rad}$

Hence for an angle $x^\circ$

$$\boxed{x^\circ=x\times\frac{\pi}{180} \text{ rad}}$$

Now lets convert Radians back into Degrees!

We know that: $1^\circ=\frac{\pi}{180} \text{ rad}$

So, we have:

$1\text{ rad}=\frac{180}{\pi}^\circ$

Hence for an angle of $x$ radians:

$$\boxed{x\text{ rad}=x\times\frac{180}{\pi}^\circ}$$

Why Radians?

Radians are very useful when you learn calculus. When you differentiate trig functions such as $\sin$, $\cos$ and $\tan$, the functions will be evaluated in Radians.

Also, Taylor-Maclaurin expansions of the trig functions above are also to be evaluated in Radians.

Degrees

Angles are measured in degrees. For example, a right angle is 90°. A right angle is 1/4 of a circle, or a full rotation, which is 90°x4=360°.

Radians

One radian is what we get when we take the radius of the circle and wrap it around the circle.

In this diagram, r is the length of the radius of the circle. The arc r is the same measure as radius r. Thus, the angle formed by the two radii is one radian, or 57.2958...°.

A fancy way to say this is: Arc r subtends the angle with measure 57.2958...°.

Let's look at the bigger picture.

If arc r subtends the angle that measures one radian, then think about what angle, in radians, the circumference of the circle subtends.

The circumference of the circle is equal to 2$\pi$r. If radians are the angle measures that arcs subtend, then the measure of the circumference is 2$\pi$ radians. This means that half a circle, or 180°, is $\pi$ radians.

180/$\pi$=57.2958...°!

We now can convert between radians and degrees!

Conversions

180° = $\pi$ radians

90°= $\pi$/2 radians

270° = 3$\pi$/2 radians

To convert an angle in degrees to radians, we must multiply by $\pi$/180.

360°x $\pi$/180 = 2$\pi$

And to convert from an angle in radians to degrees, we flip $\pi$/180 to get 180/$\pi$.

2$\pi$x180/$\pi$=360°

(Video) Converting Degrees to Radians

by Brian McLogan

This video by Brian McLogan walks you through how to convert from degrees to radians.

Summary

To start, we know that the first two quadrants (i.e., all positive $y$ values) is $180 ^{\circ}$. We also know that that same area is $2\pi$ radians.

$$\frac{\pi}{180 ^{\circ}}$$

If we want to convert $30 ^{\circ}$ to radians, we are going to multiply it by the ratio and simplify:

$$30 ^{\circ} \cdot \frac{\pi}{180 ^{\circ}} = \frac{30\pi}{180}= \frac{\pi}{6} \text{ radians}$$

This works because the degrees will cancel out, leaving us with just radians.

Now we know that $30 ^{\circ} = \frac{\pi}{6} \text{ radians}$

(Video) Converting Radians to Degrees

by Brian McLogan

This second video by Brian McLogan walks through an example of how to convert from radians to degrees.

Summary

We know that quadrants I and II are $180 ^{\circ}$ or $2\pi \text{ radians}$. We can use the ratio $\frac{180 ^{\circ}}{\pi \text{radians}}$ to convert from radians to degrees because the radian units will cancel out.

$$\frac{5\pi}{4} \cdot \frac{180 ^{\circ}}{\pi \text{radians}} = \frac{5 \cdot 180 ^{\cdot}}{4} = 22.5 ^{\cdot}$$

Using this ratio, we found that $\frac{5\pi}{4}$ radians is $22.5 ^{\cdot}$.

Converting Between Radians and Degrees

Our next step is to learn how to convert between radians and degrees.

If we want to convert from radians $\rightarrow$ degrees,

$$\text{degrees} = \text{radians} \cdot \frac{180^{\circ}}{\pi}$$

To go the other way, if we want to convert from degrees $\rightarrow$ radians,

$$\text{radians} = \text{degrees} \cdot \frac{\pi}{180^{\circ}}$$