Lesson 3.4

Coterminal Angles in Radians

Just as adding to a degree angle produces a coterminal partner, adding to a radian angle does the same. The terminal side lands in the exact same place.

Introduction

In Lesson 1.9 you found coterminal angles by adding or subtracting . The concept is identical in radians — the only difference is that a full rotation is now instead of .

Past Knowledge

You found coterminal angles in degrees by adding or subtracting .

Today's Goal

Find positive and negative coterminal angles by adding or subtracting .

Future Success

Coterminal angles are the foundation for understanding periodicity — the repeating nature of every trig function.

Key Concepts

The Coterminal Formula (Radians)

Two angles are coterminal if they share the same terminal side. To find coterminal angles in radians:

: add → one full rotation counterclockwise
: subtract → one full rotation clockwise
: add → two full rotations counterclockwise

Finding a Positive Coterminal for a Negative Angle

If given a negative radian angle, keep adding until the result is positive (between and ).

Finding a Negative Coterminal for a Positive Angle

If given a positive radian angle, subtract until the result is negative (between and ).

Common Denominator Trick

Since , you often need to rewrite it with a matching denominator. For instance, if your angle is , convert to before adding or subtracting.

Worked Examples

Basic

Finding a Positive Coterminal

Question: Find a positive coterminal angle for .

Step 1: Add .

Rewrite with denominator : .

Step 2: Compute.

Final Answer: is coterminal with .

Intermediate

Converting a Negative to Positive

Question: Find the positive coterminal angle for that lies between and .

Step 1: Add (with denominator 3).

Step 2: Check. Is between and ? Yes!

Final Answer: is the positive coterminal angle.

Advanced

Large Radian Value

Question: Find the coterminal angle of that lies between and .

Step 1: Note that . So subtract :

Step 2: Still greater than . Subtract again:

Step 3: Check. ✓

Final Answer: (we subtracted twice, meaning the original angle went around the circle full times plus ).

Common Pitfalls

Adding 360 Instead of 2π

If the angle is in radians, you must add or subtract , not . Adding to a radian angle would give you a number far beyond any reasonable angle measurement.

Forgetting to Match Denominators

You cannot add without rewriting as first. Failing to find a common denominator is the #1 arithmetic mistake in this topic.

Real-Life Applications

Clock Mechanisms

A clock's hour hand completes a full rotation ( radians) every hours. After hours, the hand has traveled radians total, but its position is coterminal with — right back where it started. This is exactly why clocks don't need to count past : coterminal angles ensure the same positions repeat forever.