🔄 Lesson 2: The Unit Circle & Radians

Angular measure, periodic functions, and the bridge to waves

🎯 Learning Objectives

• Convert between degrees and radians fluently
• Define sin, cos, tan for any angle using the unit circle
• Understand reference angles and signs in all quadrants
• Connect angular measure to angular velocity (ω) and period (T)

🔑 Key Concepts

Radians: The Natural Angle Unit

1 radian = angle subtended by an arc length equal to the radius.

Full circle: 2π rad = 360° → 1 rad ≈ 57.3°

Conversion: radians = degrees × π/180, degrees = radians × 180/π

Degrees	Radians	Common Use
30°	π/6	π/6, π/3, π/2, π, 2π
45°	π/4
60°	π/3
90°	π/2
180°	π
360°	2π

The Unit Circle Definition

For any angle θ (measured counterclockwise from +x axis):

• cos θ = x-coordinate of point on unit circle
• sin θ = y-coordinate of point on unit circle
• tan θ = sin θ / cos θ = y/x

This extends trig to all angles, not just 0°–90°!

Quadrant Signs (ASTC — All Students Take Calculus)

Quadrant	Angle Range	sin	cos	tan
I	0 to π/2	+	+	+
II	π/2 to π	+	−	−
III	π to 3π/2	−	−	+
IV	3π/2 to 2π	−	+	−

Reference Angle

The acute angle to the x-axis. sin, cos, tan have same absolute value as reference angle — sign depends on quadrant.

• QI: θ_ref = θ
• QII: θ_ref = π − θ
• QIII: θ_ref = θ − π
• QIV: θ_ref = 2π − θ

Physics Connection: Angular Velocity & Period

ω (angular velocity) = Δθ/Δt — measured in rad/s!

Period T = 2π/ω. Frequency f = 1/T = ω/2π.

📐 Worked Examples

Example 1: Degree ↔ Radian Conversion

(a) 60° → 60 × π/180 = π/3 rad
(b) 5π/6 rad → (5π/6) × 180/π = 150°
(c) 1.5 rad → 1.5 × 180/π ≈ 85.9°
            

Example 2: Unit Circle Values

sin(2π/3) = sin(π - π/3) = sin(π/3) = √3/2  (QII, sin +)
cos(2π/3) = -cos(π/3) = -½                  (QII, cos -)
tan(5π/4) = tan(π + π/4) = tan(π/4) = 1     (QIII, tan +)
sin(-π/6) = -sin(π/6) = -½                  (negative angle = clockwise)
            

Example 3: Angular Velocity

A wheel rotates at 120 rpm. Find ω in rad/s and period T.

ω = 120 rev/min × 2π rad/rev × 1 min/60 s = 4π rad/s ≈ 12.6 rad/s
T = 2π/ω = 2π/(4π) = 0.5 s
f = 1/T = 2 Hz
            

🏋️ Practice Problems

1. Convert: 90°, 135°, 270°, -45° to radians
2. Convert: π/4, 3π/2, 5π/3, 2.5 rad to degrees
3. Find sin, cos, tan: 5π/6, 7π/4, 4π/3, -π/3
4. ω = 10 rad/s. Find T, f. How many degrees in 0.1 s?
5. sin θ = -½, θ in QIII. Find θ (rad), cos θ, tan θ.
6. A point on a circle radius 2m moves at 6 m/s. Find ω, T.
7. x = 5 cos(2t). What are A, ω, T, f? (t in seconds)

✅ Answers

Click to reveal answers

1) π/2, 3π/4, 3π/2, -π/4 2) 45°, 270°, 300°, 143.2° 3) sin=½, cos=-√3/2, tan=-1/√3; sin=-√2/2, cos=√2/2, tan=-1; sin=-√3/2, cos=-½, tan=√3; sin=-√3/2, cos=½, tan=-√3

4) T=0.628s, f=1.59Hz, 57.3° 5) θ=7π/6, cos=-√3/2, tan=1/√3 6) ω=3rad/s, T=2.09s 7) A=5, ω=2rad/s, T=π≈3.14s, f=1/π≈0.318Hz

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