Math127 Graphs Of Other Trigonometric

Graphing Trigonometric Functions

Overview

• Today's focus: Graphing secant, cosecant, cotangent, tangent functions.

• Connection to previous lesson: Designed cosine graph.

Cosecant and Secant

Cosecant Function

• Definition: Cosecant is the inverse of sine.

• Graphing Steps:

  • Start by graphing the sine function.

  • From the sine graph, determine the cosecant graph.

Example: ( y = 2 \times \csc(2x) )

• Step 1: Graph ( 2 \sin(2x) )

  • Amplitude = |2| = 2

  • Period = ( \frac{2\pi}{2} = \pi )

  • Key Points: Period divided by four gives points at ( \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi )

• Step 2: Identify Vertical Asymptotes

  • Occur at x-intercepts of sine graph (where sine = 0).

Secant Function

Definition

• Definition: Secant is the inverse of cosine.

• Graphing Steps:

  • Start by graphing the cosine function.

  • Determine the secant graph based on cosine:

    • Vertical asymptotes occur where the cosine touches the x-axis.

Example: ( y = -3 \cos\left(\frac{x}{2}\right) )

• Step 1: Graph ( -3 \cos\left(\frac{x}{2}\right) )

  • Amplitude = |-3| = 3

  • Period = ( \frac{2\pi}{\frac{1}{2}} = 4\pi )

  • Key Points: Start at 0 and find points ( 0, \pi, 2\pi, 3\pi, 4\pi )

• Step 2: Identify Vertical Asymptotes

  • Asymptotes at the points where the cosine graph intersects the x-axis.

Tangent and Cotangent Functions

Information

• Tangent and cotangent graphs exhibit similar shapes but are opposites.

• Graphing Techniques:

  • Method 1: Use formulas for vertical asymptotes.

  • Method 2: Use the standard tangent/cotangent graphs to identify key points.

Vertical Asymptotes for Tangent

• Asymptotes occur at ( \frac{\pi}{2} + n\pi ) where n is an integer.

Example: ( y = \tan\left(x + \frac{\pi}{4}\right) )

• Step 1: Analyze and find vertical asymptotes ( -\frac{\pi}{4} ) and ( \frac{3\pi}{4} ).

• Step 2: Combine with identified key points to graph.

Example: ( y = -\tan\left(x - \frac{\pi}{2}\right) )

• Step 1: Find asymptotes through solving for x so that ( x - \frac{\pi}{2} = n\pi ).

• Step 2: Graph asymptotes with the cotangent function starting from the adjusted points according to change in sign.

Further Examples

Example: ( y = 3 \cot\left(\frac{1}{2}x\right) )

• Step 1: Identify asymptotes from original cotangent graph and adjust based on ( \frac{1}{2} ).

• Step 2: Graph the cotangent value and vertical asymptotes located at 0, ( \pi ), and follow standard pattern.

Final Thoughts

• Graphing techniques: Use either the formula approach or table of values.

• Systematic approach yields accurate representations for all functions.

• Practice applying concepts through different examples for mastery.