Radians and Degrees — ACT Math Guide

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

Radians and Degrees ACT