Chapter 4.6, Problem 29E

Solution

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

The values are $a=3.61,b=2$ and $h=0.49$ and the function is $f\left(x\right)\approx 3.61\sin \left(2\left(x-0.49\right)\right)$ .

Given information :

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

Explanation :

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

The graph of the function is:

  

Graph indicates that the function is a sinusoid function.

The maximum value of the function is approximately $3.61$ . Thus the amplitude of the function is $3.61$ .

The $x-$ intercept of the function near to zero is, approximately $0.49$ . Therefore, the phase shift of the function is approximately $0.49$ .

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, $a=3.61,b=2$

Also,

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

Hence, $h=0.49$

Therefore, the function is $f\left(x\right)\approx 3.61\sin \left(2\left(x-0.49\right)\right)$ .