The Foundation of Circular Measurement

What is an Angle?

Definition: An angle is formed by rotating a ray (called the terminal side) from an initial position (the initial side) about a fixed point (the vertex).

Two Perspectives:

1. Static: Two rays sharing a common endpoint

2. Dynamic: Amount of rotation from initial to terminal side

Two Systems of Angle Measurement

Degrees (°)

• One complete revolution = 360°

• Historical origin: Ancient Babylonian base-60 system

• Subdivisions: 1° = 60 minutes, 1′ = 60 seconds

Common Degree Measures:

• Right angle: 90°

• Straight angle: 180°

• Full rotation: 360°

• Acute: 0° < θ < 90°

• Obtuse: 90° < θ < 180°

Radians (rad)

• Natural measure based on arc length

• One radian = angle subtended by an arc equal in length to the radius

• One complete revolution = 2π radians (≈ 6.283 rad)

Why Radians?

• Simplifies calculus formulas

• Natural for circular motion

• Connects linear and angular measures

• AP Precalculus and Calculus prefer radians

The Critical Conversion Formula

$$\pi$$radian = $$\ang{180}$$

Conversion Formulas:

• Degrees to radians: Multiply by $$\frac{\pi}{\ang{180}}$$​

  radians=degrees×$$\frac{\pi}{\ang{180}}$$​

• Radians to degrees: Multiply by $$\frac{\ang{180}}{\pi}$$​

  degrees=radians×$$\frac{\ang{180}}{\pi}$$​

Common Angle Conversions

Coterminal Angles

Definition: Angles that share the same terminal side.

Finding Coterminal Angles:

• Add or subtract multiples of 360° (or 2π radians)

Formula:
For degrees: θ±360k° where k is any integer
For radians: θ±2πk where k is any integer

Example : Find two coterminal angles with 50°

1. 50°+360°=410°

2. 50°−360°=−310°

Reference Angles

Definition: The acute angle (0° to 90° or 0 to π/2) formed between the terminal side and the x-axis.

How to Find:

1. Determine which quadrant θ is in

2. Use these formulas:

   • Quadrant I: ref angle = θ

   • Quadrant II: ref angle = 180° – θ (or π – θ)

   • Quadrant III: ref angle = θ – 180° (or θ – π)

   • Quadrant IV: ref angle = 360° – θ (or 2π – θ)

Example 3: Find reference angle for 210°

1. 210° is in QIII (180° < 210° < 270°)

2. ref angle = 210° – 180° = 30°

Why Important? Reference angles help find trig values for any angle using acute angle values.

Arc Length Formula

When θ is measured in radians:

s=rθ

where:

• $s$ = arc length

• $r$ = radius

• $\theta$ = central angle (in radians)

If θ is in degrees: Must convert to radians first!​

Example 4: Find arc length of a circle with r = 5 cm, θ = 2 rad
s=5×2=10cm

The Unit Circle Connection

Definition: Circle with radius 1 centered at origin

• Coordinates of points: (cos θ, sin θ)

• Arc length = θ (when θ in radians)

• Circumference = 2π

Why radians work perfectly: On unit circle, arc length = angle measure in radians

Special Angles in Standard Position

Quadrantal Angles: Angles whose terminal side lies on an axis

• 0° (0 rad) = positive x-axis

• 90° (π/2) = positive y-axis

• 180° (π) = negative x-axis

• 270° (3π/2) = negative y-axis

• 360° (2π) = back to positive x-axis

XII. Common AP Exam Questions

Type 1: Conversion Practice

“Convert 225° to radians in terms of π”

Type 2: Coterminal Angles

“Find an angle between 0° and 360° coterminal with 750°”

Type 3: Reference Angles

“What is the reference angle for $$\frac{7\pi}{6}$$?”

Type 4: Application Problems

“A ferris wheel with radius 40 ft rotates at 2 rad/min. How far does a rider travel in 5 minutes?”

Type 5: Multiple Representations

“Given θ = 5π/3, sketch in standard position, give a coterminal angle, and find reference angle”