Chapter 4.6, Problem 30E

Solution

To determine : The value of $a,b,\text{ and }h$ in $f\left(x\right)=\cos 3x+2\sin 3x$ so that $f\left(x\right)\approx a\sin \left(b\left(x-h\right)\right)$ .

The values are $a=2.22,b=3$ and $h=-0.17$ and $f\left(x\right)\approx 2.22\sin \left(3\left(x+0.17\right)\right)$ .

Given information :

The function $f\left(x\right)=\cos 3x+2\sin 3x$

Explanation :

Consider the function: $f\left(x\right)=\cos 3x+2\sin 3x$ .

The graph of the function is:

  

Graph indicates that the function is a sinusoid function.

The maximum value of the function is approximately $2.22$ . Thus the amplitude of the function is $2.22$ .

The $x-$ intercept of the function near to zero is, approximately $-0.17$ . Therefore, the phase shift of the function is approximately $-0.17$ .

Consider the function of the form,

Consider the function of the form,

  $f\left(x\right)\approx a\sin \left(b\left(x-h\right)\right)$

Here, $a$ is the amplitude of the function.

Hence, $a=2.22,b=3$

Also,

Here, $h$ is the phase shift of the function.

Hence, $h=-0.17$

Therefore, the function is $f\left(x\right)\approx 2.22\sin \left(3\left(x+0.17\right)\right)$ .