Degrees and Radians

What is Radian Measure?

A radian is the angle formed when the arc length equals the radius of a circle.

Key insight: One complete revolution around a circle is $2\pi$ radians (or $360°$).

The Fundamental Conversion

$180° = \pi \text{ radians}$

This is the most important relationship to remember!

Converting Between Degrees and Radians

Degrees to Radians

Formula: Multiply by $\frac{\pi}{180}$

Example 1: Convert $45°$ to radians

$45° \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4}$

Example 2: Convert $120°$ to radians

$120° \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3}$

Radians to Degrees

Formula: Multiply by $\frac{180}{\pi}$

Example 1: Convert $\frac{\pi}{6}$ to degrees

$\frac{\pi}{6} \times \frac{180}{\pi} = \frac{180}{6} = 30°$

Example 2: Convert $\frac{5\pi}{4}$ to degrees

$\frac{5\pi}{4} \times \frac{180}{\pi} = \frac{5 \times 180}{4} = \frac{900}{4} = 225°$

Common Angle Conversions

Memorize these frequently used conversions:

| Degrees | Radians | Notes | |---------|---------|-------| | $0°$ | $0$ | Starting point | | $30°$ | $\frac{\pi}{6}$ | Half of $60°$ | | $45°$ | $\frac{\pi}{4}$ | Quarter turn | | $60°$ | $\frac{\pi}{3}$ | Twice $30°$ | | $90°$ | $\frac{\pi}{2}$ | Right angle | | $120°$ | $\frac{2\pi}{3}$ | Twice $60°$ | | $135°$ | $\frac{3\pi}{4}$ | Three times $45°$ | | $150°$ | $\frac{5\pi}{6}$ | Five times $30°$ | | $180°$ | $\pi$ | Straight angle | | $270°$ | $\frac{3\pi}{2}$ | Three-quarter turn | | $360°$ | $2\pi$ | Full circle |

Why Use Radians?

1. Simpler formulas: Arc length is just $s = r\theta$ (when $\theta$ is in radians)
2. Calculus: Derivatives and integrals of trig functions work cleanly with radians
3. Natural measure: Radians measure angles based on the circle itself, not an arbitrary division into 360 parts

Quick Tips

To convert degrees to radians mentally:

1. Divide the degree measure by $180$
2. Multiply by $\pi$
3. Simplify the fraction

Example: $270°$

• Step 1: $270 \div 180 = \frac{270}{180} = \frac{3}{2}$
• Step 2: $\frac{3}{2} \times \pi = \frac{3\pi}{2}$

Practice Problems

1. Convert $210°$ to radians
2. Convert $\frac{7\pi}{6}$ to degrees
3. Convert $315°$ to radians
4. What angle in radians is one-third of a full rotation?

Real-World Applications

• Astronomy: Angles between celestial objects
• Engineering: Rotational motion and angular velocity
• Computer graphics: Rotation transformations
• Physics: Circular motion and wave properties