Negative and Coterminal Angles

Negative Angles

What are Negative Angles?

Negative angles are measured clockwise from the positive x-axis, instead of counterclockwise.

• Positive angles: Rotate counterclockwise (standard direction)
• Negative angles: Rotate clockwise

Examples

Example 1: $-90°$ (or $-\frac{\pi}{2}$ radians)

• This is a $90°$ rotation clockwise
• Ends up pointing straight down (negative y-axis)
• Same terminal side as $270°$ (or $\frac{3\pi}{2}$)

Example 2: $-\frac{\pi}{4}$

• This is $45°$ clockwise
• Ends up in Quadrant IV
• Same terminal side as $\frac{7\pi}{4}$ or $315°$

Example 3: $-180°$

• Half circle clockwise
• Ends up on negative x-axis
• Same terminal side as $180°$

Converting Between Positive and Negative Angles

To convert a negative angle to its positive equivalent:

Add $360°$ (or $2\pi$ radians)

Examples:

Convert $-45°$ to positive: $-45° + 360° = 315°$

Convert $-\frac{\pi}{3}$ to positive: $-\frac{\pi}{3} + 2\pi = -\frac{\pi}{3} + \frac{6\pi}{3} = \frac{5\pi}{3}$

Convert $-120°$ to positive: $-120° + 360° = 240°$

Why Use Negative Angles?

Negative angles are useful for:

• Physics: Describing clockwise rotation (gears, wheels)
• Navigation: Turning right vs. left
• Computer graphics: Rotation transformations
• General math: Sometimes a negative angle is simpler to describe

Coterminal Angles

What are Coterminal Angles?

Coterminal angles are angles that have the same terminal side (they end up pointing in the same direction).

Key insight: You can make coterminal angles by adding or subtracting full rotations ($360°$ or $2\pi$ radians).

Finding Coterminal Angles

Formula: $\theta_{\text{coterminal}} = \theta + 360°n \quad \text{(or } \theta + 2\pi n \text{ in radians)}$

where $n$ is any integer ($...,-2, -1, 0, 1, 2, ...$)

Examples

Example 1: Find coterminal angles with $30°$

Add/subtract $360°$:

• $30° + 360° = 390°$ ✓
• $30° - 360° = -330°$ ✓
• $30° + 720° = 750°$ ✓
• $30° - 720° = -690°$ ✓

All of these angles are coterminal with $30°$!

Example 2: Find coterminal angles with $\frac{\pi}{6}$

Add/subtract $2\pi$:

• $\frac{\pi}{6} + 2\pi = \frac{\pi}{6} + \frac{12\pi}{6} = \frac{13\pi}{6}$ ✓
• $\frac{\pi}{6} - 2\pi = \frac{\pi}{6} - \frac{12\pi}{6} = -\frac{11\pi}{6}$ ✓

Example 3: Are $45°$ and $405°$ coterminal?

Check: $405° - 45° = 360°$

Yes! They differ by exactly one full rotation, so they're coterminal.

Example 4: Find the coterminal angle between $0°$ and $360°$ for $-210°$

Add $360°$: $-210° + 360° = 150°$

So $-210°$ and $150°$ are coterminal.

Example 5: Find the coterminal angle between $0$ and $2\pi$ for $\frac{13\pi}{4}$

Subtract $2\pi$ until we're in range: $\frac{13\pi}{4} - 2\pi = \frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4}$

So $\frac{13\pi}{4}$ and $\frac{5\pi}{4}$ are coterminal.

Standard Position

The standard position coterminal angle is the coterminal angle between:

• $0°$ and $360°$ (or $0$ and $2\pi$ in radians)

This is often the most useful form.

To find it:

1. If the angle is negative, keep adding $360°$ (or $2\pi$) until it's positive
2. If the angle is greater than $360°$ (or $2\pi$), keep subtracting $360°$ (or $2\pi$) until it's in range

Trig Values of Coterminal Angles

Important property: Coterminal angles have the same trig values!

Since they have the same terminal side on the unit circle: $\sin(\theta) = \sin(\theta + 360°n)$ $\cos(\theta) = \cos(\theta + 360°n)$ $\tan(\theta) = \tan(\theta + 360°n)$

Example: $\sin(30°) = \sin(390°) = \sin(-330°) = \frac{1}{2}$

All three angles are coterminal, so they all have the same sine value!

Practice Problems

Problem 1: Convert $-\frac{2\pi}{3}$ to a positive angle.

Problem 2: Find three coterminal angles with $120°$ (one positive, one negative, one greater than $360°$).

Problem 3: Find the standard position coterminal angle for $940°$.

Problem 4: Are $-\frac{\pi}{4}$ and $\frac{7\pi}{4}$ coterminal?

Problem 5: If $\cos(50°) = x$, what is $\cos(410°)$?

Problem 6: Find the standard position coterminal angle for $-\frac{17\pi}{6}$.

Real-World Applications

Wheels and Gears

• Rotating clockwise vs. counterclockwise
• Multiple full rotations

Navigation

• Ship/plane headings that go past $360°$
• Turning more than one full circle

Physics

• Angular displacement (can be positive or negative)
• Periodic motion (repeating angles)

Astronomy

• Planetary orbits (many full rotations)
• Celestial coordinate systems

Summary

Negative Angles:

• Measured clockwise
• Convert to positive by adding $360°$ (or $2\pi$)

Coterminal Angles:

• Share the same terminal side
• Found by adding/subtracting $360°$ (or $2\pi$)
• Have the same trig values
• Standard position: between $0°$ and $360°$ (or $0$ and $2\pi$)

Key Formula: $\theta_{\text{coterminal}} = \theta \pm 360°n \quad (\text{or } \theta \pm 2\pi n)$