Radians & Unit Circle Basics

TipLearning Objectives
  • Convert between degrees and radians.
  • Understand radian measure as arc length on the unit circle.
  • Locate key angles on the unit circle.
  • Use \(\pi\)-based reasoning for angle measures.

Key Ideas

A radian measures an angle by how much arc length it cuts off on a circle.

On the unit circle (radius 1): \[ \text{radians} = \text{arc length} \]

Degree–radian conversions:

  • \(180^\circ = \pi\) radians
  • \(1^\circ = \frac{\pi}{180}\) radians
  • \(1\text{ rad} = \frac{180}{\pi}^\circ\)

Key unit circle angles:

Degrees Radians
\(30^\circ\) \(\pi/6\)
\(45^\circ\) \(\pi/4\)
\(60^\circ\) \(\pi/3\)
\(90^\circ\) \(\pi/2\)

Unit circle with key angles labeled in degrees and radians, with special angles color-coded.

Common Problem Types

Converting Degrees ↔︎ Radians

Multiply by \(\frac{\pi}{180}\) or \(\frac{180}{\pi}\).

Identifying Quadrants

Example: \(5\pi/6\) lies in Quadrant II.

Using Arc Length = Radius × Angle

On unit circle, arc length = angle in radians.

Recognizing Equivalent Angles

E.g., \(2\pi + \theta\) is coterminal.

Strategies

  • Always reduce fractions in radian form.
  • Use \(\pi\) to visualize angle size (e.g., \(\pi/2\) = quarter-turn).
  • Convert to degrees if stuck estimating.

Worked Examples

Example 1

Convert \(120^\circ\) to radians: \[ 120^\circ \cdot \frac{\pi}{180} = \frac{2\pi}{3} \]

Example 2

Convert \(\frac{5\pi}{4}\) to degrees: \[ \frac{5\pi}{4} \cdot \frac{180}{\pi} = 225^\circ \]

WarningCommon Mistakes
  • Forgetting to include \(\pi\) in radian answers.
  • Mixing degree and radian symbols.
  • Placing angles in wrong quadrants.

Practice Problems

  1. Convert \(210^\circ\) to radians.
  2. Convert \(7\pi/6\) to degrees.
  3. What quadrant is \(3\pi/4\) in?
  1. \(210\cdot\pi/180 = 7\pi/6\)
  2. \((7\pi/6)\cdot 180/\pi = 210^\circ\)
  3. Quadrant II

Summary

  • Radians measure arc length on the unit circle.
  • Use \(\pi/180\) conversions.
  • Know benchmark radian angles.
  • Think: \(\pi\) = 180°, \(\pi/2\) = 90°, \(\pi/3\) = 60°, etc.
  • Radians grow with arc length.