Sine Calculator
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Understanding the Sine Function
What is Sine?
The sine function (sin θ) is one of the fundamental trigonometric functions. In a right triangle:
sin θ = opposite / hypotenuse
Unit Circle and Sine
On the unit circle, sine represents the y-coordinate of any point on the circle. This helps us understand why:
- Sine values are always between -1 and 1
- Sine is periodic with period 2π
- Sine of negative angles equals negative sine of positive angles
Important Properties
Domain
All real numbers (−∞ to +∞)
Range
[−1, 1]
Period
2π radians (360°)
Odd Function
sin(−x) = −sin(x)
Special Angles
| Angle (degrees) | Angle (radians) | Sine Value | Exact Form |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 | 0.5 | 1/2 |
| 45° | π/4 | 0.707 | 1/√2 |
| 60° | π/3 | 0.866 | √3/2 |
| 90° | π/2 | 1 | 1 |
Key Relationships
Pythagorean Identity
sin²θ + cos²θ = 1
Double Angle Formula
sin(2θ) = 2sin(θ)cos(θ)
Power Reduction
sin²θ = (1 - cos(2θ))/2
Real-World Applications
Physics
Used in wave motion, simple harmonic motion, and oscillations
Engineering
Essential in signal processing and electrical engineering
Architecture
Used in calculating angles and heights of structures