Answer:
Explanation:
The sine and cosine functions are periodic functions that are commonly used in mathematics and physics. These functions are defined in terms of the ratios of the sides of a right triangle, and they have a number of important properties and applications.
The sine function, denoted by sin(x), is a function that describes the ratio of the side opposite a given angle in a right triangle to the hypotenuse of the triangle. The cosine function, denoted by cos(x), is a function that describes the ratio of the adjacent side of a given angle in a right triangle to the hypotenuse of the triangle.
Both the sine and cosine functions have a domain of all real numbers and a range of all real numbers between -1 and 1. These functions are periodic, meaning that they repeat over a fixed interval known as the period. The period of the sine function is 2π, and the period of the cosine function is also 2π.
The sine and cosine functions can be graphed on the coordinate plane using the x-axis as the domain and the y-axis as the range. The sine function has a shape that is similar to a wave, with the curve oscillating between maximum and minimum values. The cosine function has a shape that is similar to a wave, but it has a phase shift of π/2 compared to the sine function. This means that the cosine function reaches its maximum and minimum values one quarter of a period later than the sine function.
To graph a sine or cosine function, you can start by plotting the intercepts, which are the points where the curve crosses the x-axis. You can then use the period and the maximum and minimum values of the function to plot additional points on the curve.
As an example, consider the sine function y = sin(x). This function has a period of 2π, which means that it repeats every 2π units on the x-axis. The maximum value of the function is 1, and the minimum value is -1. To graph this function, we can start by plotting the intercepts at x = 0 and x = 2π. We can then plot additional points on the curve using the period and the maximum and minimum values of the function. For example, at x = π, the value of the function is 0, which is the midpoint between the maximum and minimum values. At x = 3π/2, the value of the function is -1, which is the minimum value.
As a example, consider the cosine function y = cos(x). This function also has a period of 2π and maximum and minimum values of 1 and -1. To graph this function, we can start by plotting