Angle Measures
📌 Introduction to Angle Measures
Angles can be measured in two units: degrees and radians. Understanding both systems and how to convert between them is fundamental to trigonometry and advanced mathematics.
Converting Between Radians and Degrees
Key Relationship:
\( \pi \text{ radians} = 180° \)
This fundamental relationship is the basis for all conversions
Conversion Formulas:
Radians to Degrees:
\( \text{Degrees} = \text{Radians} \times \frac{180°}{\pi} \)
Degrees to Radians:
\( \text{Radians} = \text{Degrees} \times \frac{\pi}{180°} \)
Important Values:
- \( 1 \text{ radian} = \frac{180°}{\pi} \approx 57.296° \)
- \( 1° = \frac{\pi}{180} \approx 0.01745 \text{ radians} \)
- \( 2\pi \text{ radians} = 360° \) (full circle)
- \( \frac{\pi}{2} \text{ radians} = 90° \)
- \( \frac{\pi}{3} \text{ radians} = 60° \)
- \( \frac{\pi}{4} \text{ radians} = 45° \)
- \( \frac{\pi}{6} \text{ radians} = 30° \)
📝 Examples - Conversions:
Example 1: Convert \( \frac{3\pi}{4} \) radians to degrees
\( \frac{3\pi}{4} \times \frac{180°}{\pi} = \frac{3 \times 180°}{4} = \frac{540°}{4} = 135° \)
Example 2: Convert 210° to radians
\( 210° \times \frac{\pi}{180°} = \frac{210\pi}{180} = \frac{7\pi}{6} \text{ radians} \)
Example 3: Convert \( -\frac{\pi}{3} \) radians to degrees
\( -\frac{\pi}{3} \times \frac{180°}{\pi} = -\frac{180°}{3} = -60° \)
Radians and Arc Length
Arc Length Formula:
The arc length \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in radians):
\( s = r\theta \)
⚠️ Important: \( \theta \) MUST be in radians for this formula!
Definition of One Radian:
One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
- When \( \theta = 1 \) radian, then \( s = r \)
- This means the arc length equals the radius
- This is true for any circle regardless of size
📝 Examples - Arc Length:
Example 1: Find the arc length of a circle with radius 10 cm and central angle \( \frac{\pi}{3} \) radians
\( s = r\theta = 10 \times \frac{\pi}{3} = \frac{10\pi}{3} \approx 10.47 \text{ cm} \)
Example 2: A circle has radius 8 inches. Find the arc length for a central angle of 2.5 radians
\( s = 8 \times 2.5 = 20 \text{ inches} \)
Example 3: Find the central angle (in radians) if arc length is 15 cm and radius is 5 cm
\( s = r\theta \) → \( 15 = 5\theta \) → \( \theta = 3 \text{ radians} \)
Graphs of Angles in Standard Position
Standard Position:
An angle is in standard position when:
- Its vertex is at the origin
- Its initial side lies along the positive x-axis
- Its terminal side rotates from the initial side
Direction of Rotation:
Positive Angles:
Measured counterclockwise from the positive x-axis
Negative Angles:
Measured clockwise from the positive x-axis
Quadrants
The Four Quadrants:
The coordinate plane is divided into four quadrants by the x-axis and y-axis:
| Quadrant | Angle Range (Degrees) | Angle Range (Radians) | Signs |
|---|---|---|---|
| I | 0° to 90° | \( 0 \) to \( \frac{\pi}{2} \) | \( x > 0, y > 0 \) |
| II | 90° to 180° | \( \frac{\pi}{2} \) to \( \pi \) | \( x < 0, y > 0 \) |
| III | 180° to 270° | \( \pi \) to \( \frac{3\pi}{2} \) | \( x < 0, y < 0 \) |
| IV | 270° to 360° | \( \frac{3\pi}{2} \) to \( 2\pi \) | \( x > 0, y < 0 \) |
Quadrantal Angles:
Angles whose terminal sides lie on an axis (not in a quadrant):
- 0° or \( 0 \) radians (positive x-axis)
- 90° or \( \frac{\pi}{2} \) radians (positive y-axis)
- 180° or \( \pi \) radians (negative x-axis)
- 270° or \( \frac{3\pi}{2} \) radians (negative y-axis)
Coterminal Angles
Definition:
Coterminal angles are angles in standard position that have the same terminal side.
Finding Coterminal Angles:
In Degrees:
\( \theta + 360°n \) where \( n \) is any integer
In Radians:
\( \theta + 2\pi n \) where \( n \) is any integer
Add or subtract full rotations (360° or \( 2\pi \)) to find coterminal angles
📝 Examples - Coterminal Angles:
Example 1: Find a positive and negative coterminal angle to 45°
Positive: \( 45° + 360° = 405° \)
Negative: \( 45° - 360° = -315° \)
Example 2: Find the angle between 0° and 360° coterminal with 840°
\( 840° - 360° = 480° \) (still > 360°)
\( 480° - 360° = 120° \)
Answer: 120°
Example 3: Find coterminal angle to \( \frac{5\pi}{6} \) between 0 and \( 2\pi \)
\( \frac{5\pi}{6} + 2\pi = \frac{5\pi}{6} + \frac{12\pi}{6} = \frac{17\pi}{6} \) (> \( 2\pi \))
\( \frac{5\pi}{6} - 2\pi = \frac{5\pi}{6} - \frac{12\pi}{6} = -\frac{7\pi}{6} \) (negative)
The angle \( \frac{5\pi}{6} \) is already between 0 and \( 2\pi \)
Reference Angles
Definition:
A reference angle is the acute angle (smallest positive angle) formed between the terminal side of the given angle and the x-axis.
- Reference angles are always positive
- Reference angles are always acute (between 0° and 90°)
- Measured from terminal side to the x-axis (not y-axis)
Finding Reference Angles:
Step 1: Find coterminal angle between 0° and 360° (or 0 and \( 2\pi \))
Step 2: Use the appropriate formula based on quadrant:
| Quadrant | Reference Angle (Degrees) | Reference Angle (Radians) |
|---|---|---|
| I | \( \theta' = \theta \) | \( \theta' = \theta \) |
| II | \( \theta' = 180° - \theta \) | \( \theta' = \pi - \theta \) |
| III | \( \theta' = \theta - 180° \) | \( \theta' = \theta - \pi \) |
| IV | \( \theta' = 360° - \theta \) | \( \theta' = 2\pi - \theta \) |
📝 Examples - Reference Angles:
Example 1: Find the reference angle for 150°
150° is in Quadrant II
\( \theta' = 180° - 150° = 30° \)
Example 2: Find the reference angle for 225°
225° is in Quadrant III
\( \theta' = 225° - 180° = 45° \)
Example 3: Find the reference angle for \( \frac{5\pi}{3} \) radians
\( \frac{5\pi}{3} \) is in Quadrant IV
\( \theta' = 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3} \)
Example 4: Find the reference angle for -210°
First find coterminal: \( -210° + 360° = 150° \)
150° is in Quadrant II
\( \theta' = 180° - 150° = 30° \)
⚡ Quick Summary
| Concept | Formula/Key Idea |
|---|---|
| Radians to Degrees | \( \text{rad} \times \frac{180°}{\pi} \) |
| Degrees to Radians | \( \text{deg} \times \frac{\pi}{180°} \) |
| Arc Length | \( s = r\theta \) (θ in radians) |
| Coterminal Angles | \( \theta \pm 360°n \) or \( \theta \pm 2\pi n \) |
| Reference Angle | Acute angle to x-axis |
- \( \pi \) radians = 180° is the fundamental conversion relationship
- Arc length formula requires angle in radians
- Positive angles rotate counterclockwise, negative clockwise
- Coterminal angles differ by full rotations (360° or \( 2\pi \))
- Reference angles are always acute and positive
- Quadrant determines how to calculate reference angle
📚 Common Angle Conversions
| Degrees | Radians | Degrees | Radians |
|---|---|---|---|
| 0° | 0 | 180° | \( \pi \) |
| 30° | \( \frac{\pi}{6} \) | 210° | \( \frac{7\pi}{6} \) |
| 45° | \( \frac{\pi}{4} \) | 225° | \( \frac{5\pi}{4} \) |
| 60° | \( \frac{\pi}{3} \) | 240° | \( \frac{4\pi}{3} \) |
| 90° | \( \frac{\pi}{2} \) | 270° | \( \frac{3\pi}{2} \) |
| 120° | \( \frac{2\pi}{3} \) | 300° | \( \frac{5\pi}{3} \) |
| 135° | \( \frac{3\pi}{4} \) | 315° | \( \frac{7\pi}{4} \) |
| 150° | \( \frac{5\pi}{6} \) | 330° | \( \frac{11\pi}{6} \) |