AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.2A — Radians
Notes — A Natural Unit for Angle Measure
💡 Learning Objectives (3.2.A Part 1)
By the end of this lesson you will be able to:
- Define a radian as a ratio of arc length to radius
- Convert between degrees and radians
- Identify common angles in radian form (π/6, π/4, π/3, π/2, …)
- Place angles on the unit circle in standard position
1. Where Do Radians Come From?
Degrees are a historical choice — 360 is a convenient number with many divisors, but it has no mathematical meaning. Radians, by contrast, are defined directly by circles themselves:
One radian is the angle at the center of a circle that sweeps out an arc equal in length to the radius.
In any circle of radius r, an angle of θ radians sweeps out an arc of length s = r · θ. That formula is the whole reason radians are worth learning — it makes arc length and angle interchangeable up to a factor of r.
2. Converting Between Degrees and Radians
A full revolution is 360° on the degree scale. It is 2π radians on the radian scale (because the full circumference 2πr divided by the radius r gives 2π). That gives us the conversion keys:
- 180° = π rad
- 1° = π/180 rad ≈ 0.01745 rad
- 1 rad = 180/π° ≈ 57.296°
📘 Example — Conversions
- 90° × (π/180) = π/2 rad
- 60° × (π/180) = π/3 rad
- (5π/6) rad × (180/π) = 150°
- 2 rad × (180/π) ≈ 114.59°
3. Common Angles You Should Know Cold
Degrees | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
|---|---|---|---|---|---|---|---|---|---|
Radians | 0 | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π |
Degrees | 210° | 225° | 240° | 270° | 300° | 315° | 330° | 360° |
|---|---|---|---|---|---|---|---|---|
Radians | 7π/6 | 5π/4 | 4π/3 | 3π/2 | 5π/3 | 7π/4 | 11π/6 | 2π |
Memorize the first row; the rest can be derived as ‘π minus the reference’, ‘π plus the reference’, or ‘2π minus the reference.’
4. Standard Position and the Unit Circle
An angle in STANDARD POSITION has its vertex at the origin and its initial ray along the positive x-axis. A positive angle opens counter-clockwise; a negative angle opens clockwise. The point where the terminal ray intersects the unit circle (the circle of radius 1 centered at the origin) is the geometric home of every trig value we will compute.
📝 Why radians matter for calculus
In calculus, the derivative formula d/dx[sin(x)] = cos(x) only works when x is in radians. In degrees the formula carries an extra factor of π/180. Radians are the ‘natural’ angle unit — use them unless a problem explicitly says degrees.
5. Coterminal and Reference Angles
Two angles are COTERMINAL if they share a terminal ray. Adding or subtracting full revolutions (2π radians, or 360°) produces a coterminal angle. So 5π/2 is coterminal with π/2, and −π/3 is coterminal with 5π/3.
The REFERENCE ANGLE of θ is the acute angle between the terminal ray and the x-axis. It is always between 0 and π/2 and is used to compute trig values in any quadrant.
📘 Example — Reference angles
- θ = 5π/6 lies in Quadrant II; reference angle = π − 5π/6 = π/6
- θ = 4π/3 lies in Quadrant III; reference angle = 4π/3 − π = π/3
- θ = 7π/4 lies in Quadrant IV; reference angle = 2π − 7π/4 = π/4
6. Arc Length Revisited
Because s = r · θ, radians make arc length trivial. A 3-radian sweep on a circle of radius 4 m traces 12 m of arc. A quarter-circle on the unit circle traces π/2 units, because radius = 1 and θ = π/2.
⚠️ Common mistake
The formula s = r · θ REQUIRES θ in radians. Using degrees gives you a meaningless number. If you only have a degree measure, convert first.
7. Summary
- A radian is the angle whose arc length equals the radius
- 180° = π rad; multiply by π/180 to convert to radians, 180/π to convert back
- Memorize the unit-circle angles in multiples of π/6 and π/4
- Coterminal angles differ by multiples of 2π; reference angles are the acute angle to the x-axis
- s = rθ works directly in radians and is the key formula for circular geometry