Radians are a unit of angle measurement, similar to degrees. They offer a way to measure angles based on the radius of a circle. One radian is defined as the angle at the centre of a circle, where the length of the arc is equal to the radius of the circle.

Comparison with Degrees

• Relation with Degrees: π radians are equivalent to 180°. This means that a full circle, which is 360°, is equal to 2π radians.
• Conversion Formulas:
  • To convert radians to degrees: Multiply by $\frac{180}{\pi}$.
  • To convert degrees to radians: Multiply by $\frac{\pi}{180}$.

Image courtesy of Mometrix

Practical Examples

Example 1: Converting Radians to Degrees

Convert $\frac{3\pi}{4}$ radians into degrees.

Solution:

$ \frac{3\pi}{4} = \frac{3\pi}{4} \times \frac{180}{\pi} = 3 \times \frac{180}{4} = 135° $<h3><strong>Example 2: Converting Degrees to Radians</strong></h3><p>Convert 45° into radians.</p><p><strong>Solution:</strong></p>$45° = 45 \times \frac{\pi}{180} = \frac{\pi}{4}$<h2 id="practice-exercises"><strong>Practice Exercises</strong></h2><h4><strong>1. Convert 60° to Radians</strong></h4><p><strong>Solution:</strong></p><p>To convert degrees to radians, use the formula$\text{radians} = \text{degrees} \times \frac{\pi}{180}$.</p>$60° = 60 \times \frac{\pi}{180} = \frac{60}{180} \times \pi = \frac{1}{3} \times \pi = \frac{\pi}{3}$<p>Therefore, 60° is equal to$\frac{\pi}{3}$radians.</p><h4></h4><h4><strong>2. Convert </strong>$(\frac{\pi}{3})$<strong> Radians to Degrees</strong></h4><p><strong>Solution:</strong></p><p>To convert radians to degrees, use the formula$\text{degrees} = \text{radians} \times \frac{180}{\pi}.$</p>$\frac{\pi}{3} = \frac{\pi}{3} \times \frac{180}{\pi} = \frac{1}{3} \times 180 = 60°$<p>Therefore,$\frac{\pi}{3}$radians is equal to 60°.</p><h4></h4><h4><strong>3. Convert 120° to Radians</strong></h4><p><strong>Solution:</strong></p><p>Use the same formula for converting degrees to radians.</p>$120° = 120 \times \frac{\pi}{180} = \frac{120}{180} \times \pi = \frac{2}{3} \times \pi = \frac{2\pi}{3}$<p>Therefore, 120° is equal to$\frac{2\pi}{3}$radians.</p><h4></h4><h4><strong>4. Convert </strong>$\frac{5\pi}{6}$<strong> Radians to Degrees</strong></h4><p><strong>Solution:</strong></p><p>Again, use the formula for converting radians to degrees.</p>$\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5}{6} \times 180 = 150°$<p>Therefore,$\frac{5\pi}{6}$ radians is equal to 150°.