TR-19: Graphing Tangent and Cotangent (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial will guide you through the process of graphing the tangent and cotangent functions, as covered in the video by Dennis F. Davis. Understanding these functions is essential in trigonometry, especially when dealing with periodic behavior and asymptotes. By the end of this guide, you will be able to graph these functions accurately and identify their key characteristics.

Step 1: Understand the Basic Functions

Before graphing, it's important to know the basic definitions and properties of the tangent and cotangent functions.

• Tangent Function (tan x):

  • Defined as the ratio of the sine and cosine functions:
    • tan x = sin x / cos x
  • It has a period of π (180 degrees).

• Cotangent Function (cot x):

  • Defined as the reciprocal of the tangent function:
    • cot x = cos x / sin x
  • It also has a period of π.

Step 2: Identify Asymptotes

Asymptotes are critical for graphing tangent and cotangent functions. They occur where the function is undefined.

• For the tangent function:

  • Asymptotes are found where cos x = 0.
  • This occurs at:
    • x = (π/2) + nπ for n ∈ Z (where n is any integer).

• For the cotangent function:

  • Asymptotes are found where sin x = 0.
  • This occurs at:
    • x = nπ for n ∈ Z.

Step 3: Plot Key Points

To create an accurate graph, plot key points within one period of the function.

• Tangent:

  • Key points to consider:
    • (0, 0)
    • (π/4, 1)
    • (π/2, undefined)
    • (3π/4, -1)
    • (π, 0)

• Cotangent:

  • Key points to consider:
    • (0, undefined)
    • (π/4, 1)
    • (π/2, 0)
    • (3π/4, -1)
    • (π, undefined)

Step 4: Sketch the Graph

Using the key points and asymptotes, sketch the graph of each function.

• For Tangent:

  • Start from a point where the function is defined (e.g., (0, 0)).
  • As you approach the asymptotes, the graph will rise towards infinity or fall towards negative infinity.
  • Repeat this pattern for the next period.

• For Cotangent:

  • Start at the asymptote (e.g., (0, undefined)).
  • The graph will decrease from the asymptote to the next key point, (π/2, 0).
  • Sketch the curve approaching the asymptote as it moves towards the next interval.

Step 5: Analyze the Graphs

After sketching, analyze the graphs for patterns and properties.

• Both functions are periodic with a period of π.
• The tangent function has a range of all real numbers, while cotangent shares this property but has different key points and asymptotes.
• Identify the behavior at asymptotes and how the function approaches these points.

Conclusion

In this tutorial, you learned how to graph the tangent and cotangent functions, identify their asymptotes, plot key points, and analyze the resulting graphs. Practicing these steps will help solidify your understanding of trigonometric functions. For further study, explore other trigonometric functions and their properties, or revisit the video for additional insights. Happy graphing!