Angles in Trigonometry: Degrees, Radians, and Rotation

Angles as Rotation: Initial Side, Terminal Side, and Direction

In trigonometry, an angle is best understood as a rotation. Imagine a ray (a half-line) starting from a fixed point (the vertex). One ray stays fixed as the initial side, and a second ray rotates to become the terminal side.

Standard position is the most common setup: the vertex is at the origin, and the initial side lies on the positive x-axis. The terminal side is wherever the rotation ends.

• Positive rotation: counterclockwise (CCW).
• Negative rotation: clockwise (CW).

Think of a clock hand: moving it CCW is positive, moving it CW is negative.

How to visualize the rotation (diagram description)

Picture coordinate axes. Draw a ray along the positive x-axis (initial side). Now rotate another ray from that x-axis to some new position (terminal side). The curved arrow between them shows the direction and amount of rotation.

• If the terminal side ends in Quadrant I (upper right), the angle is between 0 and 90 degrees (or 0 and π/2 radians) in standard position.
• If it ends in Quadrant II (upper left), it is between 90 and 180 degrees (or π/2 and π radians).
• If it ends in Quadrant III (lower left), it is between 180 and 270 degrees (or π and 3π/2 radians).
• If it ends in Quadrant IV (lower right), it is between 270 and 360 degrees (or 3π/2 and 2π radians).

Degrees: The Familiar Rotation Measure

Degrees divide a full rotation into 360 equal parts.

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• Full turn: 360°
• Half turn: 180°
• Quarter turn: 90°

Degrees are convenient for describing everyday turns, bearings, and simple angle sizes.

Radians: The Natural Angle Measure for Trigonometry

Radians measure angles by comparing an arc length to a radius. This makes radians tightly connected to circles and to the formulas of trigonometry and calculus.

Arc length and the radian definition

Take a circle of radius r. If an angle θ (in radians) cuts off an arc of length s, then:

s = rθ

This formula is not a “special trick”; it is the definition of radians in action.

Why radians are useful

• Direct link to the circle: θ tells you how many “radius-lengths” fit along the arc.
• Cleaner formulas: many trig relationships and advanced formulas assume θ is in radians.
• Unit circle simplicity: when r = 1, the arc length equals the angle measure: s = θ.

Key fact: 2π radians is one full rotation

On a circle of radius r, the full circumference is 2πr. Using s = rθ for a full turn:

2πr = rθ  ⇒  θ = 2π

So:

• 360° = 2π radians
• 180° = π radians
• 90° = π/2 radians

Using s = rθ in Practical Problems

Be careful: the formula s = rθ requires θ in radians.

Example 1: Find arc length

A circle has radius r = 5 cm. An angle of θ = 1.2 radians intercepts an arc. Find s.

Step-by-step:

• Use s = rθ
• s = 5(1.2) = 6

The arc length is 6 cm.

Example 2: Find the angle in radians from arc length

A circle has radius r = 10 m. An arc length is s = 7 m. Find θ.

Step-by-step:

• Start with s = rθ
• Divide both sides by r: θ = s/r
• θ = 7/10 = 0.7

The angle is 0.7 radians.

Converting Between Degrees and Radians

Use the equivalence 180° = π radians. Conversions are just multiplying by a “unit fraction” that cancels the old unit.

Degrees to radians

Multiply by π/180.

θ (radians) = θ (degrees) × (π/180)

Example 1: Convert 60° to radians

Step-by-step:

• Start: 60°
• Multiply by π/180: 60 × (π/180)
• Simplify 60/180 = 1/3

60° = π/3

Example 2: Convert 225° to radians

Step-by-step:

• 225 × (π/180)
• Simplify 225/180 by dividing by 45: 225/180 = 5/4

225° = 5π/4

Radians to degrees

Multiply by 180/π.

θ (degrees) = θ (radians) × (180/π)

Example 3: Convert 3π/5 to degrees

Step-by-step:

• (3π/5) × (180/π)
• Cancel π
• Compute (3/5) × 180 = 108

3π/5 = 108°

Example 4: Convert −7π/6 to degrees

Step-by-step:

• (−7π/6) × (180/π)
• Cancel π
• Compute −7 × 30 = −210

−7π/6 = −210°

Coterminal Angles: Same Terminal Side, Different Rotations

Two angles are coterminal if they end at the same terminal side (same final ray), even if they rotate different amounts to get there.

In degrees, coterminal angles differ by multiples of 360°:

θ_coterminal = θ + 360°k

In radians, coterminal angles differ by multiples of 2π:

θ_coterminal = θ + 2πk

where k is any integer (..., −2, −1, 0, 1, 2, ...).

Number line view (wrapping around)

Imagine a number line of angle measures. Every time you add 360° (or 2π), you have completed one full wrap around the circle and land on the same direction again. This is why coterminal angles repeat regularly.

Circular rotation view (visual description)

Picture the unit circle with a terminal side pointing at some direction. If you rotate one extra full turn and stop, the terminal side points to the same direction. That “extra loop” changes the angle measure but not the final ray.

Example 1: Find two coterminal angles for 40°

Step-by-step:

• Add 360°: 40° + 360° = 400°
• Subtract 360°: 40° − 360° = −320°

So 400° and −320° are coterminal with 40°.

Example 2: Find a coterminal angle between 0° and 360° for −150°

Step-by-step:

• Add 360° once: −150° + 360° = 210°
• 210° is between 0° and 360°

The coterminal angle is 210° (Quadrant III).

Example 3: Find a coterminal angle between 0 and 2π for −5π/3

Step-by-step:

• Add 2π: −5π/3 + 2π = −5π/3 + 6π/3 = π/3
• π/3 is between 0 and 2π

The coterminal angle is π/3 (Quadrant I).

Positive/Negative Rotation and Quadrant Location

To identify quadrant location in standard position:

• First, find a coterminal angle in a standard range: 0° to 360° or 0 to 2π.
• Then, locate which interval it falls in (Quadrant I, II, III, IV) or whether it lands exactly on an axis.

Example: Determine quadrant for −420°

Step-by-step:

• Add 360°: −420° + 360° = −60° (still negative)
• Add 360° again: −60° + 360° = 300°
• 300° is between 270° and 360°

So the terminal side is in Quadrant IV, and the original rotation is negative (clockwise).

Mini-Quiz: Quick Checks

Part A: Conversions

• 1) Convert 30° to radians.
• 2) Convert 150° to radians.
• 3) Convert 7π/4 to degrees.
• 4) Convert −2π/3 to degrees.

Part B: Rotation direction and quadrant

• 5) Is −135° a positive or negative rotation? What quadrant is its terminal side in?
• 6) Find a coterminal angle between 0° and 360° for 765°. Then state the quadrant.
• 7) Find a coterminal angle between 0 and 2π for −11π/6. Then state the quadrant.
• 8) An angle is 3.5 radians. Is it between π and 3π/2, between 3π/2 and 2π, or between 0 and π? (Use π ≈ 3.14.)

Answer key (brief)

• 1) π/6
• 2) 5π/6
• 3) 315°
• 4) −120°
• 5) Negative; coterminal 225° → Quadrant III
• 6) 765° − 720° = 45° → Quadrant I
• 7) −11π/6 + 2π = π/6 → Quadrant I
• 8) 3.5 is slightly bigger than π (≈3.14) and less than 3π/2 (≈4.71), so it is between π and 3π/2