The sine function, written as sin(x), is a fundamental trigonometric function. It produces a smooth wave that oscillates between negative one and positive one. The function starts at zero, reaches its maximum of one at pi over two, returns to zero at pi, reaches its minimum of negative one at three pi over two, and completes one full cycle at two pi.
On the unit circle, the sine function has a clear geometric interpretation. As we move a point around the circle, the angle x is measured from the positive x-axis. The sine of this angle equals the y-coordinate of the point on the circle. Watch how the sine value changes as the point moves around the circle.
The sine function has several key values that are important to remember. At zero, sine equals zero. At pi over two, sine reaches its maximum value of one. At pi, sine returns to zero. At three pi over two, sine reaches its minimum value of negative one. Finally, at two pi, sine completes one full cycle and returns to zero again.
The sine function has several important properties. It has a period of two pi, meaning the pattern repeats every two pi units. The range is from negative one to positive one, so sine values are always bounded. The domain includes all real numbers. Sine is an odd function, which means sin of negative x equals negative sin of x. The function is also continuous and smooth everywhere.
The sine function has numerous practical applications across many fields. It describes wave motion and oscillations in physics, models sound and light waves, represents alternating current in electronics, and describes pendulum motion. In engineering, it's used for signal processing, computer graphics, and animation. The general form of a sine wave is y equals A sine of omega t plus phi, where A is amplitude, omega is frequency, and phi is phase shift. For example, y equals 2 sine of 3x plus pi over 4 represents a wave with amplitude 2, frequency 3, and phase shift pi over 4.