Chapter 4.7, Problem 96E

To determine

Write the function in terms of the sine function by using the identity

  $A\cos \omega t+B\sin \omega t=\sqrt{A^2+B^2}\sin \left(\omega t+\arctan \frac{A}{B}\right)$ and graph both forms of the function, to find the graph imply.

Answer to Problem 96E

  $f\left(t\right)=5\sin \left(\pi t+\arctan \left(\frac{4}{3}\right)\right)$

Explanation of Solution

Given information:

Write the function in terms of the sine function by using the identity

  $A\cos \omega t+B\sin \omega t=\sqrt{A^2+B^2}\sin \left(\omega t+\arctan \frac{A}{B}\right)$

Given function

  $f\left(t\right)=4\cos \pi t+3\sin \pi t$ .

Calculation:

We have the expression as $f\left(t\right)=4\cos \pi t+3\sin \pi t$ ……..(1)

Using identity $A\cos \omega t+B\sin \omega t=\sqrt{A^2+B^2}\sin \left(\omega t+\arctan \frac{A}{B}\right)$

(1)becomes

  $f\left(t\right)=5\sin \left(\pi t+\arctan \left(\frac{4}{3}\right)\right)$ ……………(2)

Use Ti-83 calculator to graph (1) and (2)

Steps for graph (1)

Step 1: Start Ti-83 calculator and press $\left[Y=\right]$ and input the following function

  $Y_1=4\cos \left(\pi X\right)+3\sin \left(\pi X\right)$

The display is as shown below

  

Step 2: Press the GRAPH button to get the graph as shown below

  

Steps for graph of (2)

Step 1: Start Ti-83 calculator and press $\left[Y=\right]$ and input the following function

  $Y_1=5\sin \left(\pi x+{\tan }^{-1}\left(4/3\right)\right)$

The display is as shown below

  

Step 2: Press the GRAPH button to get the graph as shown below

  

As both the graphs overlap on each other, this implies that both the forms are same and the identity is correct.