📐 Introduction to Radians
In trigonometry, radian measure is an alternative to degree measure
for angles. It is based on the length of the arc of a circle.
Definition:
One radian is the angle at the centre of a circle which is
subtended by an arc equal in length to the radius of the circle.
\[
1 \text{ radian} = \frac{\text{Arc length}}{\text{Radius}}
\]
Conversion between degrees and radians:
Since the circumference of a circle is \( 2\pi r \) and a full revolution is \( 360^\circ \),
\[
360^\circ = 2\pi \text{ radians}.
\]
Therefore,
\[
1 \text{ radian} = \frac{180^\circ}{\pi}, \qquad
1^\circ = \frac{\pi}{180} \text{ radians}.
\]
| Degrees (°) |
Radians |
| 0° | 0 |
| 30° | \(\displaystyle \frac{\pi}{6}\) |
| 45° | \(\displaystyle \frac{\pi}{4}\) |
| 60° | \(\displaystyle \frac{\pi}{3}\) |
| 90° | \(\displaystyle \frac{\pi}{2}\) |
| 180° | \(\displaystyle \pi\) |
| 360° | \(\displaystyle 2\pi\) |
📝 Examples
Example 1: Convert \( 120^\circ \) to radians.
\[
120^\circ \times \frac{\pi}{180^\circ}
= \frac{120\pi}{180}
= \frac{2\pi}{3}\ \text{radians}.
\]
Example 2: Convert \( \frac{3\pi}{4} \) radians to degrees.
\[
\frac{3\pi}{4} \times \frac{180^\circ}{\pi}
= \frac{3\times 180^\circ}{4}
= 135^\circ.
\]
Example 3: How many radians are there in \( 1 \) full revolution?
One revolution = \( 360^\circ \).
\[
360^\circ \times \frac{\pi}{180^\circ} = 2\pi \ \text{radians}.
\]
📌 Key Points:
- Radians are a natural unit for measuring angles in mathematics and physics.
- \(\pi\) radians \(=\) \(180^\circ\).
- Radian measure simplifies many trigonometric formulas (e.g., derivatives, arc length).