Sine Calculator
Calculate the sine (sin) of any angle or find angles from sine values with step-by-step explanation
Calculate Sine Function
Enter the angle for which to calculate sine
Click to set common angle values
Sine Results
Formula: sin(x) = opposite/hypotenuse
Input angle: 0° (0.0000 rad)
Range: -1 ≤ sin(x) ≤ 1
Function Analysis
Step-by-Step Calculation
Step 1: Convert to Standard Form
Given angle: 0°
Convert to radians: 0° × (π/180) = 0.000000 rad
Step 2: Calculate Sine
sin(0.000000) = 0.000000
Exact value: 0
Step 3: Unit Circle Interpretation
On the unit circle, sine represents the y-coordinate
sin(0°) = y-coordinate = 0.000000
Sine Function Properties
Domain
All real numbers (-∞, +∞)
No restrictions on input angles
Range
[-1, 1]
-1 ≤ sin(x) ≤ 1
Period
2π radians (360°)
sin(x + 2π) = sin(x)
Symmetry
Odd function
sin(-x) = -sin(x)
Special Angle Values
Understanding the Sine Function
What is the Sine Function?
The sine function is one of the fundamental trigonometric functions. In a right triangle, sine represents the ratio of the length of the opposite side to the hypotenuse. In the unit circle, sine represents the y-coordinate of a point on the circle.
Key Characteristics
- •sin(x) = opposite/hypotenuse in right triangles
- •sin(x) = y-coordinate on unit circle
- •Range is always between -1 and 1
- •Periodic with period 2π (360°)
Mathematical Relationships
Right Triangle
sin(θ) = opposite/hypotenuse
Unit Circle
sin(θ) = y-coordinate
Pythagorean Identity
sin²(x) + cos²(x) = 1
Note: The sine function is periodic and continuous everywhere. It oscillates smoothly between -1 and 1, creating the characteristic sine wave pattern.