Understanding the Sine Function
The sine function is one of the fundamental trigonometric functions. In a right triangle, sin(x) is defined as the ratio of the length of the side opposite to angle x to the length of the hypotenuse. On the unit circle, sin(x) gives the y-coordinate of the point at angle x.
Common Sine Values
sin(0°) = 0
The sine of 0 degrees is 0.
sin(0) = 0
sin(30°) = 0.5
One of the most commonly used values in trigonometry.
sin(π/6) = 1/2
sin(45°) = √2/2
Approximately 0.7071.
sin(π/4) = √2/2
sin(60°) = √3/2
Approximately 0.8660.
sin(π/3) = √3/2
sin(90°) = 1
The maximum value of the sine function.
sin(π/2) = 1
sin(180°) = 0
Sine returns to zero at half a full rotation.
sin(π) = 0
Properties of Sine
- Domain: all real numbers.
- Range: [-1, 1].
- Period: 2π radians (360°).
- Odd function: sin(-x) = -sin(x).
- Derivative: d/dx sin(x) = cos(x).
- Integral: ∫ sin(x) dx = -cos(x) + C.
Sine in Real Life
The sine function appears in many real-world contexts including sound waves, light waves, alternating current (AC) electricity, pendulum motion, and seasonal temperature variations. It is essential in physics, engineering, music, and signal processing.