Degrees and radians are two ways to measure angles. Degrees divide a full turn into \(360\) equal parts,
while radians measure angle using arc length on a circle.
1. Degrees
In degree measure, one full revolution is:
\[
360^\circ
\]
Common degree angles include \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\).
2. Radians
In radian measure, one full revolution is:
\[
2\pi\text{ radians}
\]
Therefore:
\[
180^\circ=\pi\text{ radians}
\]
This is the key relationship used for all degree-radian conversions.
3. Convert degrees to radians
To convert degrees to radians, multiply by \(\pi/180\):
\[
\theta_{\text{rad}}
=
\theta^\circ\cdot\frac{\pi}{180}
\]
Example:
\[
150^\circ
=
150\cdot\frac{\pi}{180}
=
\frac{5\pi}{6}
\]
4. Convert radians to degrees
To convert radians to degrees, multiply by \(180/\pi\):
\[
\theta^\circ
=
\theta_{\text{rad}}\cdot\frac{180}{\pi}
\]
Example:
\[
\frac{5\pi}{6}\cdot\frac{180}{\pi}
=
150^\circ
\]
5. Arc length
Radians are useful because they connect angle directly to arc length:
\[
s=r\theta
\]
In this formula, \(\theta\) must be in radians. If an angle is given in degrees, convert it to radians first.
6. Unit-circle coordinates
On the unit circle, an angle \(\theta\) corresponds to the point:
\[
(\cos\theta,\sin\theta)
\]
For common angles, these coordinates often have exact forms. For example:
\[
150^\circ=\frac{5\pi}{6}
\]
\[
(\cos150^\circ,\sin150^\circ)
=
\left(-\frac{\sqrt3}{2},\frac12\right)
\]
7. Formula summary
| Concept |
Formula |
Meaning |
| Full turn |
\(360^\circ=2\pi\) |
Degrees and radians for one revolution |
| Half turn |
\(180^\circ=\pi\) |
Main conversion fact |
| Degrees to radians |
\(\theta_{\text{rad}}=\theta^\circ\cdot\dfrac{\pi}{180}\) |
Multiply degrees by \(\pi/180\) |
| Radians to degrees |
\(\theta^\circ=\theta_{\text{rad}}\cdot\dfrac{180}{\pi}\) |
Multiply radians by \(180/\pi\) |
| Arc length |
\(s=r\theta\) |
\(\theta\) must be in radians |
| Unit circle |
\((x,y)=(\cos\theta,\sin\theta)\) |
Coordinates of the angle point |
8. Common mistakes
- Do not use \(180/\pi\) when converting degrees to radians; use \(\pi/180\).
- Do not use \(\pi/180\) when converting radians to degrees; use \(180/\pi\).
- Always use radians in the arc length formula \(s=r\theta\).
- Remember that \(360^\circ=2\pi\), not \(\pi\).
- Angles that differ by \(360^\circ\) or \(2\pi\) are coterminal.
