Graphing Trigonometric Functions

A Comprehensive Guide to Graphing Trigonometric Functions

Trigonometric functions describe the relationship between the sides and angles of a right triangle and are fundamental mathematical concepts. To gain a thorough understanding of these functions, it is crucial to create visual representations of their graphs on a coordinate plane. This not only helps identify their significant features but also aids in analyzing the effects of these features on their appearance. In this article, we will define trigonometric function graphs, discuss their key characteristics, and provide practical examples to guide you in graphing these functions and their reciprocals.

Trigonometric Function Graphs: An Overview

Trigonometric function graphs are graphical depictions of functions or ratios based on the sides and angles of a right triangle. These functions include sine (sin), cosine (cos), tangent (tan), and their corresponding reciprocals: cosecant (csc), secant (sec), and cotangent (cot).

Important Features of Trigonometric Function Graphs

Prior to delving into the graphing of trigonometric functions, it is crucial to identify several key features, which are as follows:

Amplitude: The vertical stretch factor, referred to as the amplitude, can be calculated by taking the absolute value of half the difference between the maximum and minimum values of a function. For functions in the form of y = asin(bx), y = acos(bx), or y = atan(bx), the amplitude equals the absolute value of a. For y = atanh(bx), the amplitude is 2. The tangent function's graph does not have an amplitude since it does not have a maximum or minimum value.
Period: The distance along the x-axis from the beginning of a pattern to where it begins again is known as the period. For sine and cosine functions, the period is 2π or 360º. For functions in the form of y = asin(bx), y = acos(bx), or y = atan(bx), the horizontal stretch factor is known as b, and specific formulas can be used to calculate the period. The period can be calculated differently for y = atanh(bx). For specific formulas, please refer to this article.
Domain and Range: The domain and range of primary trigonometric functions are as follows: Note that these functions are periodic, meaning their values repeat after a specific period.

How to Graph Trigonometric Functions?

To graph trigonometric functions, follow these steps:

If the function is in the form of y = asin(bx), y = acos(bx), or y = atan(bx), determine the values of a and b. Calculate the amplitude and period using the formulas mentioned earlier.
Create a table of ordered pairs for the points that will be included in the graph. The first value of the ordered pairs will correspond to the angle θ, and the second value will represent the trigonometric function's value for that angle (e.g., (θ, sin θ)).
For commonly used angles, use the unit circle to help calculate the values of sine and cosine.
Plot several points on the coordinate plane to complete at least one period of the function.
Connect the points with a smooth and continuous curve to obtain the graph.