Section 5.1

Angles & Radian Measure

Radians are the default angle unit for the rest of this course. Arc length, sector area, the unit circle (Section 5.2), trig graphs (Chapter 6), and every identity in Chapter 7 all assume radians. Expect 2–3 conversion and arc length questions on Exam 3.

⚡ Quick Summary

You already know degrees — a full spin is 360°. Radians are just a different way to measure the same thing, like switching from miles to kilometers. The key fact is $180° = \pi$ radians. To go from degrees to radians, multiply by $\frac{\pi}{180}$ . To go back, multiply by $\frac{180}{\pi}$ . That's it — just unit conversion.

Degrees to Radians

\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}

Multiply the degree measure by π/180

Radians to Degrees

\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}

Multiply the radian measure by 180/π

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Why bother with radians? Because the formulas for arc length and sector area only work in radians. Arc length is simply $s = r\theta$ (radius times angle), and sector area is $A = \frac{1}{2}r^2\theta$ . These formulas are beautifully simple *because* of radians — if you use degrees instead, you'd need an extra $\frac{\pi}{180}$ correction factor everywhere. Radians keep the math clean.

Here's a quick way to remember which conversion makes the number bigger or smaller: radians are small numbers (like $\frac{\pi}{4}$ ) and degrees are big numbers (like 45°). So multiplying by $\frac{\pi}{180}$ shrinks your number (degrees → radians), and multiplying by $\frac{180}{\pi}$ grows it (radians → degrees).

Coterminal angles are angles that point in the exact same direction — they just take different numbers of full spins to get there. To find one, add or subtract $360°$ (or $2\pi$ in radians). For example, $-150°$ and $210°$ are coterminal because $-150° + 360° = 210°$ . They land on the same spot on the circle.

Interactive Diagram

Key Formulas

Degrees to Radians

\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}

Multiply the degree measure by π/180

Radians to Degrees

\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}

Multiply the radian measure by 180/π

Arc Length

s = r\theta

Where s is arc length, r is radius, and θ is the angle in radians

Area of a Sector

A = \frac{1}{2}r^2\theta

Only works when θ is in radians

Coterminal Angles

\theta \pm 360°n \quad \text{or} \quad \theta \pm 2\pi n \quad (n = 1, 2, 3, \ldots)

Add or subtract full rotations to find angles that land in the same position

Key Takeaways

✓To convert: degrees → radians multiply by $\frac{\pi}{180}$ ; radians → degrees multiply by $\frac{180}{\pi}$
✓Arc length $s = r\theta$ and sector area $A = \frac{1}{2}r^2\theta$ require $\theta$ in radians
✓Coterminal angles differ by full rotations: $\theta \pm 360°$ or $\theta \pm 2\pi$
✓ $180° = \pi$ radians — this single fact drives every conversion

⚠️ Common Mistakes

✗Forgetting to convert degrees to radians before using $s = r\theta$ or $A = \frac{1}{2}r^2\theta$ — these formulas only work with radians
✗Converting the wrong direction: multiplying by $\frac{\pi}{180}$ goes degrees → radians, not the other way around
✗Not simplifying the fraction after converting — always reduce (e.g., $\frac{60\pi}{180} = \frac{\pi}{3}$ , not leaving it unsimplified)
✗Confusing coterminal with supplementary: coterminal angles differ by full rotations ( $360°$ or $2\pi$ ), supplementary angles add to $180°$

Next: Section 5.2

5.2 Unit Circle — Sine & Cosine