Lesson 2: Radian Measure and Converting Between Degrees and Radians
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define a radian using the relationship between arc length and radius
- Convert degrees to radians and radians to degrees
- Recognize exact radian values for common angles
- Find coterminal angles using radian measure
What Is a Radian?
A radian is a natural way to measure angles based on the geometry of a circle rather than an arbitrary division into 360 parts.
Radian — The measure of a central angle whose intercepted arc has length equal to the radius of the circle. Equivalently, θ (in radians) = arc length / radius = s / r.
Since the circumference of a circle is C = 2πr, one full rotation corresponds to an angle of 2πr / r = 2π radians. Therefore:
- Full rotation: 360° = 2π rad
- Half rotation: 180° = π rad
- Quarter rotation: 90° = π/2 rad
The Fundamental Relationship
The key equation connecting degrees and radians is:
180° = π radians
This gives us the conversion factors:
To convert degrees → radians: multiply by π / 180
To convert radians → degrees: multiply by 180 / π
Example 1: Converting Degrees to Radians
Convert 60° to radians.
Solution:
60° × (π / 180°) = 60π / 180 = π/3 radians
Example 2: Converting Degrees to Radians
Convert 225° to radians.
Solution:
225° × (π / 180°) = 225π / 180 = 5π/4 radians
Example 3: Converting Radians to Degrees
Convert 3π/4 radians to degrees.
Solution:
(3π/4) × (180° / π) = 3(180°)/4 = 540°/4 = 135°
Common Angle Conversions
You should memorize these standard conversions:
| Degrees | Radians | Degrees | Radians |
|---|---|---|---|
| 0° | 0 | 210° | 7π/6 |
| 30° | π/6 | 225° | 5π/4 |
| 45° | π/4 | 240° | 4π/3 |
| 60° | π/3 | 270° | 3π/2 |
| 90° | π/2 | 300° | 5π/3 |
| 120° | 2π/3 | 315° | 7π/4 |
| 135° | 3π/4 | 330° | 11π/6 |
| 150° | 5π/6 | 360° | 2π |
| 180° | π |
Coterminal Angles in Radians
Just as with degrees, coterminal angles in radians differ by full rotations of 2π.
Coterminal in Radians: θ + 2πn, where n is any integer.
Example 4: Coterminal Angles in Radians
Find one positive and one negative angle coterminal with π/3.
Solution:
Positive: π/3 + 2π = π/3 + 6π/3 = 7π/3
Negative: π/3 − 2π = π/3 − 6π/3 = −5π/3
Why Radians Matter
You might wonder why we bother with radians when degrees seem simpler. Radians are essential because:
- Calculus: The derivative of sin(x) is cos(x) only when x is in radians.
- Arc length: The formula s = rθ works directly when θ is in radians.
- Physics: Angular velocity, torque, and wave equations all use radians.
- Simplicity: Many formulas become cleaner without conversion factors.
Example 5: Non-Standard Conversion
Convert 2.5 radians to degrees (round to one decimal place).
Solution:
2.5 × (180° / π) = 450° / π ≈ 143.2°
Quick Tips for Converting
Use these strategies to convert efficiently:
- Multiply by the fraction: Always set up the conversion so the unit you want to eliminate cancels out.
- Simplify before multiplying: Reduce the fraction before calculating.
- Benchmark check: π ≈ 3.14, so π/2 ≈ 1.57, π/3 ≈ 1.05, π/4 ≈ 0.79, π/6 ≈ 0.52.
Example 6: Converting with Simplification
Convert 150° to radians.
Solution:
150 × (π / 180) = 150π / 180
Simplify by dividing numerator and denominator by 30: 5π/6
Check Your Understanding
1. Convert 270° to radians.
2. Convert 7π/4 radians to degrees.
3. Find a positive coterminal angle for −2π/3.
4. Approximately how many degrees is 1 radian?
Key Takeaways
- A radian is the angle for which the arc length equals the radius: θ = s/r.
- 180° = π radians is the fundamental conversion relationship.
- Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
- Coterminal angles in radians differ by multiples of 2π.
- Radians are essential for calculus, physics, and the arc length formula.
Ready for More?
Next Lesson
In Lesson 3 you will apply radian measure to calculate arc length and area of a sector.
Start Lesson 3