Exploring the Properties and Behavior of the Sine Function: A Guide

y = sin (x)

The equation y = sin(x) represents a sine function

The equation y = sin(x) represents a sine function.

The sine function, sin(x), is a mathematical function that describes a smooth, periodic oscillation. It relates the angle x (measured in radians) to the y-coordinate of a point on the unit circle.

In the given equation y = sin(x), the variable x represents the input or independent variable, while y represents the output or dependent variable. For any given value of x, we can calculate the corresponding value of y using the sine function.

To better understand the behavior of the sine function, let’s consider some key points:

1. Period: The sine function is periodic with a period of 2π radians or 360 degrees. This means that it repeats itself after every 2π radians. For example, sin(0) = 0, sin(2π) = 0, sin(4π) = 0, and so on.

2. Amplitude: The amplitude of the sine function is 1. This means that the maximum value of sin(x) is 1 and the minimum value is -1.

3. Symmetry: The sine function is an odd function, meaning that it exhibits symmetry about the origin. This can be seen in the equation y = sin(x) as sin(-x) = -sin(x).

4. Range: The range of the sine function is [-1, 1]. The y-values of the sine function always lie between -1 and 1 inclusively.

5. Zeroes: The sine function has infinitely many x-intercepts, known as zeroes or roots. These occur at x-values where sin(x) = 0. The first such zero is at x = 0, and subsequent zeroes occur at intervals of π radians or 180 degrees.

By graphing the equation y = sin(x), you will obtain a smooth, wave-like curve that oscillates between -1 and 1 as x increases or decreases. The curve starts at the origin (0, 0) and repeats itself every 2π radians. The shape of the graph is a periodic wave that is commonly known as a sine wave.

Remember, when working with trigonometric functions like sine, it’s important to keep in mind the units of measurement (radians or degrees) and any restrictions or specific contexts determined by the problem or application at hand.

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