TR-43: Trig Graph Variations 2 (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial covers the essential aspects of the B parameter in the general trigonometric equation ( y = A \cdot \cos(B\theta + C) + D ). Understanding how the B parameter influences the period and frequency of the wave function is crucial for students studying trigonometry, especially in preparation for international A-level exams.

Step 1: Understand the General Trigonometric Equation

• Familiarize yourself with the components of the equation:
  • A: Amplitude (vertical stretch)
  • B: Frequency (affects period)
  • C: Phase shift (horizontal shift)
  • D: Vertical shift
• Recognize that the B parameter specifically affects how often the function repeats within a given interval.

Step 2: Determine the Frequency and Period

• The frequency (f) is determined by the formula: [ f = \frac{B}{2\pi} ]
• The period (T), which is the length of one complete cycle of the wave, is calculated using: [ T = \frac{2\pi}{B} ]
• Example:
  • If ( B = 2 ), then:
    • Frequency: ( f = \frac{2}{2\pi} = \frac{1}{\pi} )
    • Period: ( T = \frac{2\pi}{2} = \pi )

Step 3: Visualize Changes in the Graph

• Use graphing tools or software to visualize how changing B affects the graph of the cosine function.
• For example:
  • With ( B = 1 ): Standard period of ( 2\pi ).
  • With ( B = 2 ): Period compresses to ( \pi ), resulting in more cycles within the same interval.

Step 4: Experiment with Different Values

• Practice by substituting various values for A, B, C, and D in the equation:
  • Observe how each parameter influences the graph.
• Suggested values to try:
  • ( A = 1, B = 1, C = 0, D = 0 )
  • ( A = 2, B = 3, C = \frac{\pi}{4}, D = 1 )

Step 5: Analyze Real-World Applications

• Discuss how trigonometric functions model real-world phenomena, such as sound waves, light waves, and seasonal variations.
• Consider how changing parameters can represent different scenarios in practical applications.

Conclusion

In this tutorial, we explored the B parameter's impact on the frequency and period of the cosine function within the general trigonometric equation. By understanding these concepts, you can better analyze trigonometric graphs and their practical applications. For further study, practice graphing varied equations and consider how each parameter influences the results.