Angles are often measured in two units: degrees and radians. While degrees are more commonly used in everyday life, radians are the standard unit of angular measurement in mathematics, physics, and engineering. This guide will walk you through the simple steps of converting degrees to radians, along with examples and handy reference tables for quick conversions.
Understanding Degrees and Radians
- Degrees: A circle is divided into 360 equal parts, and each part is one degree (°). Thus, a full circle is 360°.
- Radians: Radians measure angles based on the radius of a circle. A complete circle is (2π) radians. In simple terms, 180° is equal to π radians.
The Formula for Conversion
To convert degrees to radians, you can use the following formula:
- (π) (Pi): A constant approximately equal to 3.14159.
- 180: Represents the total degrees in half a circle.
Step-by-Step Conversion Process
Step 1: Multiply the Degree Value by π
The first step is to multiply the given degree value by (π). For example, if you have 45°, multiply it by (π):
Step 2: Divide by 180
Next, divide the result by 180 to complete the conversion:
Step 3: Simplify the Fraction
If possible, simplify the resulting fraction. For 45°, the fraction simplifies to:
So, 45° = π/4 radians.
Common Degree to Radian Conversions
Here’s a table of commonly used degree values and their corresponding radian equivalents:
| Degrees (°) | Formula | Radians |
|---|---|---|
| 30° | 30 × π/180 | π/6 |
| 45° | 45 × π/180 | π/4 |
| 60° | 60 × π/180 | π/3 |
| 90° | 90 × π/180 | π/2 |
| 120° | 120 × π/180 | 2π/3 |
| 135° | 135 × π/180 | 3π/4 |
| 180° | 180 × π/180 | π |
| 270° | 270 × π/180 | 3π/2 |
| 360° | 360 × π/180 | 2π |
Examples of Degree to Radian Conversion
Example 1: Convert 120° to Radians
Use the formula:
Simplify the fraction:
Example 2: Convert 210° to Radians
Use the formula
Simplify the fraction
Example 3: Convert 75° to Radians
Use the formula:
Simplify the fraction:
Quick Reference Table for Degree to Radian Conversions
For quick conversions, use this reference table:
| Degrees (°) | Radians |
|---|---|
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Tips for Quick Conversion
- Remember Key Angles: Memorizing the radian equivalents of common angles (like 30°, 45°, 60°, 90°, and 180°) can make conversions faster.
- Use Approximate Values: When you need a numerical approximation, use (π ≈ 3.14159). For example, 45° = π/4 ≈ 0.785 radians.
- Use a Calculator: For angles not easily simplified (like 75° or 210°), using a calculator to multiply the degree by π/180 provides a quick conversion.
Practice Problems
- Convert 150° to radians.
- Use the formula: 150 x π/180
- Simplify: 5π/6
- Answer: 150° = 5π/6 radians.
- Use the formula: 150 x π/180
- Convert 300° to radians.
- Use the formula: 300 x π/180
- Simplify: 5π/3
- Answer: 300° = 5π/3 radians.
- Use the formula: 300 x π/180
Converting degrees to radians is straightforward with the formula. The key to mastering this process is practicing with various angles and familiarizing yourself with common conversions. Keep the reference tables handy for quick conversions.