Radians and Degrees — ACT Math Guide

Q: What You Need to Know

• **Degrees**: Full circle = 360°, based on ancient Babylonian math • **Radians**: Full circle = 2π radians, based on circle circumference • **Key relationship**: 180° = π radians • **Common angles**: 30°, 45°, 60°, 90° and their radian equivalents • **Calculator tip**: Your calculator can work in b

Radians and degrees ACT problems appear regularly in the trigonometry section, testing your ability to convert between these two angle measurement systems. These units measure the same thing — angles — but use different scales, like Celsius and Fahrenheit for temperature. On the ACT math section, you'll encounter 2-4 questions involving radians and degrees among the 60 questions in 60 minutes. Don't worry — once you understand the conversion relationship, these problems become straightforward point-grabbers.

What You Need to Know

Degrees: Full circle = 360°, based on ancient Babylonian math

Radians: Full circle = 2π radians, based on circle circumference

Key relationship: 180° = π radians

Common angles: 30°, 45°, 60°, 90° and their radian equivalents

Calculator tip: Your calculator can work in both degree and radian modes

📐 KEY FORMULA: To convert degrees to radians: multiply by π/180°

To convert radians to degrees: multiply by 180°/π

⏱️ ACT TIME TIP: Memorize common conversions (30°, 45°, 60°, 90°) to save time — with 60 questions in 60 minutes, every second counts!

How to Solve Radians and Degrees on the ACT

Example Question 1 — Easy/Medium Difficulty

What is 120° expressed in radians?

A) π/6

B) π/3

C) 2π/3

D) 3π/4

E) 5π/6

Solution:

Step 1: Use the conversion formula: degrees × π/180°

Step 2: Substitute: 120° × π/180°

Step 3: Simplify: 120π/180 = 2π/3

✅Answer: C — 120° equals 2π/3 radians after simplifying the fraction.

Example Question 2 — Hard Difficulty

If sin(5π/6) = 1/2, what is the value of sin(150°)?

A) -1/2

B) -√3/2

C) 1/2

D) √3/2

E) 1

Solution:

Step 1: Convert 5π/6 radians to degrees: (5π/6) × (180°/π) = 150°

Step 2: Since 5π/6 radians = 150°, both expressions represent the same angle

Step 3: Therefore, sin(5π/6) = sin(150°) = 1/2

✅Answer: C — The same angle expressed in different units gives the same sine value.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting to convert between radians and degrees when the problem mixes units

✅Fix: Always check if angles are in the same units before calculating

❌Mistake: Using the wrong conversion factor (using 360°/π instead of 180°/π)

✅Fix: Remember that π radians = 180°, not 360°

❌Mistake: Leaving answers in decimal form when the ACT wants exact radian values

✅Fix: Keep answers with π when working with radians — don't convert to decimals

❌Mistake: Not setting calculator to the correct mode (degree vs. radian)

✅Fix: Check your calculator mode before computing trig functions

Practice Question — Try It Yourself

Convert 3π/4 radians to degrees.

A) 120°

B) 135°

C) 150°

D) 180°

E) 225°

Show Answer

Answer: B — (3π/4) × (180°/π) = 540°/4 = 135°

Key Takeaways for the ACT

Master the conversion formulas: multiply by π/180° for degrees to radians, multiply by 180°/π for radians to degrees

Memorize common angle conversions to speed through ACT math questions quickly

Your calculator allows both degree and radian modes — just make sure you're in the right one

ACT math radians and degrees questions often combine with other trigonometry concepts

Practice recognizing equivalent angles in both measurement systems