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## Modeling Phenomena with Trigonometric Functions

#### 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.