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Modeling Phenomena with Trigonometric Functions

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Aligned To Common Core Standard:

High School - HSF-TF.B.5

How to Model Periodic Phenomena using Trigonometric Functions? Every motion that repeats itself is categorized under a periodic phenomenon. Circular motion, rotation of an object, movement of a pendulum, propagation of a wave, and reciprocating motion are all forms of periodic motion. If you have a clear understanding of graphing trigonometric functions, you can easily model a periodic phenomenon on a graph. The general form of cosine and sine function is; y=A sin⁡B(x-h)+K, y=cos⁡ B(x-h)+K These functions can be easily plotted to display a periodic motion of an object. Here, A is the amplitude of the sinusoidal or cosine wave. It is the height of the crest or depth of the trough. K is the value of vertical shift. If K is greater than zero, the graph will shift upwards, and if K is less than zero, the graph will shift downwards. The time it takes to one complete cycle is the period of the wave. The period can be calculated as; Period=2π/|B| h in the function determines the horizontal shift of the graph. If h is greater than zero, the graph shifts to the right, and if h is less than zero, the graph shifts to the left. These worksheets and lessons can be used to learn how to visualize phenomena through the use of trigonometry.

Printable Worksheets And Lessons

Homework Sheets

Find the period and amplitude of mixed functions.

  • Homework 1 - The period of sin (x) or cos (x) = 2π / (Multiplier of x or θ) The period of tan (x) = π / (Multiplier of x or Θ).
  • Homework 2 - Modeling of functions follow the formula: f(t) = A (B – C) + D . A is the amplitude. B is the period. C is the phase shift. D is the vertical shift.
  • Homework 3 - Evaluate each choice by running it through the paces.

Practice Worksheets

The multiple choice version was actually written into the standard original. They have since removed it.

  • Practice 1 - What is the period of the function cos (0.5Φ).
  • Practice 2 - What is the vertical shift of the function y = 8 sin (5x/14) + 9.
  • Practice 3 - What is the amplitude of the function y = 5 sin (21x/8)

Math Skill Quizzes

We add the concept of vertical shifts to our vocabulary here.

  • Quiz 1 - Students key on the skills of identifying period, amplitude, and vertical shift.
  • Quiz 2 - See if you can predict where these are going.
  • Quiz 3 - This quiz has many applications in physics.