1. What Is an Angle?
An angle is formed when one line rotates from its initial position to a new position around a fixed point (called the vertex). The amount of rotation determines the size of the angle.
To measure this rotation, two systems are commonly used:
- Degree measure
- Radian measure
2. Degree Measure
The degree is the most familiar unit for measuring angles. A complete rotation forms an angle of:
\( 360^\circ \)
This system comes from ancient Babylonian mathematics, which used base 60.
2.1. Common Degree Angles
- Right angle = 90°
- Straight line = 180°
- Full rotation = 360°
3. Radian Measure
A radian is another way to measure angles. It is widely used in higher mathematics, calculus and physics because it relates angles to arc lengths on a circle.
An angle of 1 radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius.
The full rotation (one circumference) equals:
\( 2\pi \text{ radians} \)
3.1. Key Idea Behind Radians
If an arc length is equal to the radius, the angle is 1 radian. So the relationship between arc length (\(s\)), radius (\(r\)), and angle (\(\theta\)) in radians is:
\( \theta = \dfrac{s}{r} \)
4. Relation Between Degrees and Radians
Since a complete circle is both 360° and 2π radians, we have the basic relationship:
\( 180^\circ = \pi \text{ radians} \)
From this, we derive conversion formulas:
\( 1^\circ = \dfrac{\pi}{180} \text{ radians} \)
\( 1 \text{ radian} = \dfrac{180}{\pi} ^\circ \)
4.1. Examples
1. Convert 60° to radians:
\( 60^\circ = 60 \times \dfrac{\pi}{180} = \dfrac{\pi}{3} \)
2. Convert \( \dfrac{\pi}{4} \) radians to degrees:
\( \dfrac{\pi}{4} = \dfrac{\pi}{4} \times \dfrac{180}{\pi} = 45^\circ \)
5. Most Common Angles in Both Units
Some angles are used very frequently. Here is a reference table:
| Degrees | Radians |
|---|---|
| 0° | 0 |
| 30° | \( \dfrac{\pi}{6} \) |
| 45° | \( \dfrac{\pi}{4} \) |
| 60° | \( \dfrac{\pi}{3} \) |
| 90° | \( \dfrac{\pi}{2} \) |
| 180° | \( \pi \) |
| 360° | \( 2\pi \) |
6. Why Radians Are Preferred in Higher Mathematics
Radians create simpler formulas, especially in calculus:
- \( \dfrac{d}{d\theta}(\sin \theta) = \cos \theta \)
- \( \dfrac{d}{d\theta}(\cos \theta) = -\sin \theta \)
- Length of arc:
\( s = r\theta \)
- Sector area:
\( A = \dfrac{1}{2} r^2 \theta \)