How to Solve: Radian and Degree Conversion

How to Solve: Radian and Degree Conversion

Introduction

"If you can’t convert between radians and degrees, you’ll lose marks on every trigonometry question—from unit circles to calculus. Let’s fix that in 5 minutes."

What You Need To Know First

1. Degrees: A full circle = 360° (like a clock).
2. Radians: A full circle = 2π radians (based on the radius of the circle).
3. π (pi): Approximately 3.14159, the ratio of a circle’s circumference to its diameter.

If you’re shaky on these, pause and review before continuing.

Key Vocabulary

Formulas To Know

1. Degrees to Radians

Formula: [ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) ]

Variables: - Degrees: The angle in degrees (e.g., 45°). - π/180: Conversion factor (memorise this).

Mark: MEMORISE THIS (not always given on exams).

2. Radians to Degrees

Formula: [ \text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right) ]

Variables: - Radians: The angle in radians (e.g., π/4). - 180/π: Conversion factor (memorise this).

Mark: MEMORISE THIS (not always given on exams).

3. Special Angles (Shortcut)

Memorise these common conversions (saves time on exams):

Mark: MEMORISE THIS TABLE (examiners expect you to know these).

Step-by-Step Method

Converting Degrees to Radians

1. Write down the given angle in degrees.
2. Multiply by π/180.
3. Why? Because 180° = π radians, so each degree = π/180 radians.
4. Simplify the fraction (if possible).
5. Write the final answer in radians (leave π as π).

Example: Convert 120° to radians. 1. Given: 120°. 2. Multiply: ( 120 \times \frac{\pi}{180} ). 3. Simplify: ( \frac{120\pi}{180} = \frac{2\pi}{3} ). 4. Answer: ( \frac{2\pi}{3} ) radians.

Converting Radians to Degrees

1. Write down the given angle in radians.
2. Multiply by 180/π.
3. Why? Because π radians = 180°, so each radian = 180/π degrees.
4. Simplify the fraction (if possible).
5. Write the final answer in degrees (include the ° symbol).

Example: Convert ( \frac{5\pi}{6} ) radians to degrees. 1. Given: ( \frac{5\pi}{6} ). 2. Multiply: ( \frac{5\pi}{6} \times \frac{180}{\pi} ). 3. Simplify: ( \frac{5 \times 180}{6} = 150 ). 4. Answer: 150°.

Worked Examples

Example 1 - Basic: Convert 45° to Radians

Steps: 1. Given: 45°. 2. Multiply: ( 45 \times \frac{\pi}{180} ). 3. Simplify: ( \frac{45\pi}{180} = \frac{\pi}{4} ). 4. Answer: ( \frac{\pi}{4} ) radians.

What we did and why: - We used the degrees-to-radians formula. - Simplified the fraction by dividing numerator and denominator by 45.

Example 2 - Medium: Convert ( \frac{7\pi}{4} ) Radians to Degrees

Steps: 1. Given: ( \frac{7\pi}{4} ). 2. Multiply: ( \frac{7\pi}{4} \times \frac{180}{\pi} ). 3. Simplify: ( \frac{7 \times 180}{4} = 315 ). 4. Answer: 315°.

What we did and why: - We used the radians-to-degrees formula. - Cancelled π and simplified the fraction.

Example 3 - Exam Style: Find the Radian Measure of 210°

Steps: 1. Given: 210°. 2. Multiply: ( 210 \times \frac{\pi}{180} ). 3. Simplify: ( \frac{210\pi}{180} = \frac{7\pi}{6} ). 4. Answer: ( \frac{7\pi}{6} ) radians.

What we did and why: - Recognised 210° is 30° past 180° (π radians). - Simplified the fraction by dividing numerator and denominator by 30.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Memorise the special angles—0°, 30°, 45°, 60°, 90°—and their radian equivalents. Always simplify your fractions, and never forget the units. If you see a tricky angle like 210°, break it down: 180° + 30°, so π + π/6 = 7π/6. Double-check your work for common mistakes, like mixing up the formulas or dropping π. You’ve got this—now go ace that exam!