# Modeling with Trigonometric Functions Worksheets

We often use math to help us make confident well-informed decisions. When it comes to anything that can be categorized as periodic, meaning regular and repeated, trigonometric functions are quite helpful. They help us model these phenomena in graphic form. Using these graphs, we can then predict the location of an object in the future we a great level of accuracy. Using this type of math scientists can predict the location of a location in atom. That is crazy when you understand that the speed of an electron is close 1,400 miles per hour. These worksheets and lessons can be used to learn how to visualize phenomena through the use of trigonometry.

### Aligned Standard: HSF-TF.B.5

- Period of 5 Pi Step-by-step Lesson - A period is one cycle of a wave, if you forgot.
- Guided Lesson - We calculate the amplitude and period of functions.
- Guided Lesson Explanation - If you understand the format of the equation, you should be light years ahead of other students.
- Practice Worksheet - These types of questions are commonly multiple choice. I tried to match that in this problem set.
- Matching Worksheet - This is just like a giant multiple choice set up here.

- Answer Keys - These are for all the unlocked materials above.

### 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.

### How to Model Periodic Phenomena using Trigonometric Functions

Every motion that repeats itself is categorized as a periodic phenomenon. Since it repeats itself, it becomes predictable and regular. 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. Once you have a clear graph it makes it simple to understand where these things will be at what time.

The sine and the cosine functions are the most frequently used functions to help us model the physical world. These functions can be easily plotted to display a periodic motion of an object. The general form of cosine and sine function is: y = A sinB(x-h) + K, y = cos B(x-h)+K. 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.