# 3.3: Getting Triggy With It

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 2, Lesson 6.

## Problem 1 – A general trigonometric function

Using the Transformation Graphing app, press Y=\begin{align*}Y=\end{align*} and enter the general sine function in Y1\begin{align*}Y_1\end{align*},

Y1=A  sin(B  X+C)+D.\begin{align*}Y_1 = A \ ^* \ \sin(B \ ^* \ X + C) + D.\end{align*}

Complete the table.

A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} C\begin{align*}C\end{align*} D\begin{align*}D\end{align*} zero1 zero2 min max
1 1 0 0
4 12\begin{align*}\frac{1}{2}\end{align*} 3 1

## Problem 2 – The effect of the coefficients A, B, C, and D

Examining A

• Set B=1\begin{align*}B = 1\end{align*} and C=D=0\begin{align*}C = D = 0\end{align*} and change the value of A\begin{align*}A\end{align*}. Try 4 different values of A\begin{align*}A\end{align*}.
A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} C\begin{align*}C\end{align*} D\begin{align*}D\end{align*} zero1 zero2 min max
1 0 0
1 0 0
1 0 0
1 0 0
• How did the appearance of the graph change?
• Which graph features changed? Which did not change?
• Write equations to describe the relationship between A\begin{align*}A\end{align*} and the features that did change.
• When B=1\begin{align*}B = 1\end{align*} and C=D=0\begin{align*}C = D = 0\end{align*}, ___________________.

The value of A\begin{align*}A\end{align*} is the amplitude. It is equal to half of the difference between its maximum and minimum values.

• Calculate the amplitude from the minimum and maximum values in the table above.
• Compare the results to the values of A\begin{align*}A\end{align*}. What do you notice?

Examining B

A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} C\begin{align*}C\end{align*} D\begin{align*}D\end{align*} zero1 zero2 min max
1 0 0
1 0 0
1 0 0
1 0 0
• Try 4 different values of B\begin{align*}B\end{align*}. How did the appearance of the graph change?
• Which graph features changed? Which did not change?
• Describe the relationship between B\begin{align*}B\end{align*} and the features that did change.

Examining C

A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} C\begin{align*}C\end{align*} D\begin{align*}D\end{align*} zero1 zero2 min max
1 1 0
1 1 0
1 1 0
1 1 0
• Try 4 different values of C\begin{align*}C\end{align*}. How did the appearance of the graph change?
• Which graph features changed? Which did not change?
• What is the effect of an increasing sequence of values for C\begin{align*}C\end{align*} on the graph?
• What is the effect of a decreasing sequence of values for C\begin{align*}C\end{align*} on the graph?

Examining D

A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} C\begin{align*}C\end{align*} D\begin{align*}D\end{align*} zero1 zero2 min max
1 1 0
1 1 0
1 1 0
1 1 0
• Try 4 different values of D\begin{align*}D\end{align*}. How did the appearance of the graph change?
• Try an increasing sequence of values for D\begin{align*}D\end{align*} such as 0, 1, 2, 3, 4 ... What is the effect on the graph?
• Try a decreasing sequence of values for D\begin{align*}D\end{align*} such as 0, -1, -2, -3, -4 ... What is the effect on the graph?
• Describe the effect of the value of D\begin{align*}D\end{align*} on the graph. How does changing D\begin{align*}D\end{align*} change the graph features?

## Problem 3 – A closer look at amplitude, period, and frequency

In Y1\begin{align*}Y_1\end{align*}, enter the general cosine function, A  cos(B  X+C)+D\begin{align*}A \ ^* \ \cos(B \ ^* \ X + C) + D\end{align*}.

amplitude: half of the vertical distance from minimum value to maximum value

period: horizontal distance from one peak (maximum point) to the next

frequency: number of cycles per 2π\begin{align*}2\pi\end{align*} interval

• Write a formula to find the frequency f\begin{align*}f\end{align*} given the period p\begin{align*}p\end{align*}.
• Use the formula to complete the table on the next page.
A\begin{align*}A\end{align*} B\begin{align*}B\end{align*} \begin{align*}C\end{align*} \begin{align*}D\end{align*} max point min point next max point amplitude period frequency
1 1 0 0 (0, 1) (3.14, -1) (6.28, 1)

\begin{align*}\frac{1}{2}*(1 - (-1))\end{align*}

2

\begin{align*}6.28 - 0\end{align*}

6.28

\begin{align*}2\pi\end{align*}

1 0 0
1 0 0
1 0 0
1 0 0
1 1 0
1 1 0
1 1 0
1 1 0
1 1 0
• Based on the results in the table, determine and record each relationship:
• \begin{align*}A\end{align*} and the amplitude
• \begin{align*}B\end{align*} and the frequency
• \begin{align*}B\end{align*} and the period

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