Sin Degrees Calculator
Calculate sine function for angles in degrees with exact values and comprehensive analysis
Calculate Sine in Degrees
Enter any angle in degrees (e.g., 45, 90, 180, 270, 360)
Sine Results
Exact Value
sin(0°) = 0 ≈ 0.000000
Zero angle
Angle Analysis
Input Angle: 0.000000°
Normalized Angle: 0.000000° (0° to 360°)
Quadrant: 1 (0° to 90°)
Reference Angle: 0.000000°
Radians: 0.000000 rad
Calculation Details
Formula: sin(θ) where θ is in degrees
Conversion to radians: θ (rad) = θ (°) × π/180
Calculation: sin(0.000000°) = sin(0.000000 rad) = 0.000000
Range: -1 ≤ sin(θ) ≤ 1
Period: 360° (sine repeats every 360°)
Step-by-Step Solution
- Step 1: Input angle = 0.000000°
- Step 2: Convert to radians: 0.000000° × (π/180) = 0.000000 rad
- Step 3: Normalize angle to 0°-360° range: 0.000000°
- Step 4: Determine quadrant: Quadrant 1
- Step 5: Calculate reference angle: 0.000000°
- Step 6: Apply sine function: sin(0.000000°) = 0.000000
Common Sine Values
| Angle (°) | Exact Value | Decimal | Quadrant | Description |
|---|---|---|---|---|
| 0° | 0 | 0.000000 | Axis | Zero angle |
| 30° | 1/2 | 0.500000 | I | Common angle |
| 45° | √2/2 | 0.707107 | I | Perfect square |
| 60° | √3/2 | 0.866025 | I | Common angle |
| 90° | 1 | 1.000000 | I | Maximum value |
| 120° | √3/2 | 0.866025 | II | Second quadrant |
| 135° | √2/2 | 0.707107 | II | Second quadrant |
| 150° | 1/2 | 0.500000 | II | Second quadrant |
| 180° | 0 | 0.000000 | II | Half circle |
| 210° | -1/2 | -0.500000 | III | Third quadrant |
| 225° | -√2/2 | -0.707107 | III | Third quadrant |
| 240° | -√3/2 | -0.866025 | III | Third quadrant |
| 270° | -1 | -1.000000 | III | Minimum value |
| 300° | -√3/2 | -0.866025 | IV | Fourth quadrant |
| 315° | -√2/2 | -0.707107 | IV | Fourth quadrant |
| 330° | -1/2 | -0.500000 | IV | Fourth quadrant |
| 360° | 0 | 0.000000 | Axis | Full circle |
Sine Function Properties
Range
[-1, 1]
Period
360° (2π radians)
Symmetry
Odd function: sin(-θ) = -sin(θ)
Quadrant Reference
Quadrant I (0° to 90°)
sin(θ) > 0 (positive)
Quadrant II (90° to 180°)
sin(θ) > 0 (positive)
Quadrant III (180° to 270°)
sin(θ) < 0 (negative)
Quadrant IV (270° to 360°)
sin(θ) < 0 (negative)
Calculator Tips
Use DMS format for precise angle measurements
Sine values repeat every 360°
Common angles have exact values
Check quadrant for sign determination
Understanding the Sine Function in Degrees
What is the Sine Function?
The sine function is a fundamental trigonometric function that relates an angle to the ratio of the opposite side to the hypotenuse in a right triangle. When working with degrees, we measure angles in the familiar 360-degree system where a full rotation equals 360°.
Unit Circle Definition
On the unit circle, sin(θ) represents the y-coordinate of the point where the angle θ intersects the circle. This geometric interpretation helps understand why sine values range from -1 to 1 and repeat every 360°.
Quadrant Behavior
Quadrant I (0° - 90°): Sine values from 0 to 1
Quadrant II (90° - 180°): Sine values from 1 to 0
Quadrant III (180° - 270°): Sine values from 0 to -1
Quadrant IV (270° - 360°): Sine values from -1 to 0
Special Angles
Certain angles have exact sine values that can be expressed using simple fractions and square roots. These include 30°, 45°, 60°, and their multiples, which frequently appear in mathematical problems and real-world applications.
Applications
Engineering
Structural analysis, wave calculations, and oscillatory motion modeling
Navigation
GPS calculations, bearing measurements, and celestial navigation
Physics
Wave mechanics, alternating current analysis, and harmonic motion
Mathematical Properties
Periodicity
sin(θ + 360°) = sin(θ) for any angle θ
Symmetry
sin(-θ) = -sin(θ) (odd function)
Complementary Angles
sin(90° - θ) = cos(θ)
Supplementary Angles
sin(180° - θ) = sin(θ)