The sine function, written as y equals sine of x, is one of the most important functions in mathematics. Its domain includes all real numbers, while its range is restricted to values between negative one and positive one. The function has a period of two pi, meaning it repeats its pattern every two pi units.
The sine function has a beautiful geometric interpretation using the unit circle. When we place a point on the unit circle at angle theta, the y-coordinate of that point gives us the sine of theta. As the angle changes and the point moves around the circle, we can see how the sine value oscillates between negative one and positive one.
Let's examine the key properties of the sine function. The amplitude is one, meaning the function oscillates between negative one and positive one. The period is two pi, so the pattern repeats every two pi units. The function has zeros at multiples of pi, reaches its maximum value of one at pi over two plus two n pi, and its minimum value of negative one at three pi over two plus two n pi. The sine function is also an odd function, meaning sine of negative x equals negative sine of x.
The sine function can be transformed in several ways. Multiplying by a constant A changes the amplitude, making the wave taller or shorter. Multiplying x by B changes the frequency and period. Adding a constant C to x creates a horizontal phase shift. Finally, adding a constant D shifts the entire graph vertically. Let's see these transformations in action.
To summarize what we have learned about the sine function: y equals sine of x is a fundamental trigonometric function with domain of all real numbers and range from negative one to positive one. It has a period of two pi and amplitude of one. The function has a beautiful geometric interpretation through the unit circle, and it can be transformed in various ways to model different phenomena in mathematics, physics, and engineering.