Sine Function
f(x) = sin x
The sine function is one of the six trigonometric functions that relates the angles of a right triangle to the lengths of its sides. It is denoted by the symbol sin.
In a right triangle with angle θ (measured in radians), the sine of θ is defined as the ratio of the length of the side opposite to θ to the length of the hypotenuse. Mathematically, we can represent the sine function as follows:
sin(θ) = opposite/hypotenuse
We can also represent the sine function using the unit circle, where the value of sin(θ) at any point on the unit circle is the y-coordinate of that point.
Some important properties of the sine function include:
1. It is periodic with a period of 2π radians, meaning that its values repeat every 2π radians.
2. Its range is between -1 and 1, inclusive. This means that the sine function can never exceed a value of 1 or fall below a value of -1.
3. It is an odd function, meaning that sin(-θ) = -sin(θ).
4. Its maximum and minimum values are 1 and -1, respectively, and these occur when the angle θ is equal to π/2 and 3π/2 radians, respectively.
The sine function is used extensively in mathematics and science to model a wide range of phenomena, including waves and vibrations. It is also used in engineering applications, such as in signal processing and control systems.
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