Question 1206910
.


        In part 1,  the solution by @MathLover for the shift is  INCORRECT.


        In part 2,  the final formula for sine function in the post by @MathLover1 is  INCORRECT.



                        Below is my solution for Part 1.



1. Determine the period, amplitude, vertical displacement, and phase shift for the following function.


            y = -(1/2)*cos(6𝜋x/5 + 3)-7



<pre>
In this problem, the key point is to determine correctly the phase shift 
of the given function relative to the parent function cos(x).


To apply the STANDARD procedure of analyzing the function y = -(1/2)*cos(6𝜋x/5 + 3)-7,

it should be first written with POSITIVE LEADING COEFFICIENT.


So, instead of -(1/2) at the beginning, the POSITIVE (1/2} must go.
For it, the shift in the cosine argument must be changed by adding or subtracting {{{pi}}}
to shift cosine for half of the period, because shifting half the period for cosine
is equivalent to changing the sign at cosine.


Thus, we should replace the given function by THIS ONE  y = (1/2)(cos(6𝜋x/5 + 3-𝜋).


Now we can apply the standard analyzing procedure, which gives


    - the amplitude is 1/2;

    - the period is {{{2pi/(((6pi)/5))}}} = {{{5/3}}};

    - vertical displacement, or the midline of the plot is -7;

    - phase shift is  {{{-((3-pi)/(((2pi)/5)))}}} = {{{(pi-3)/(((2pi)/5))}}}.
</pre>

It is a correct solution and the correct answer for Part 1.




In part 2, &nbsp;the correct answer for sine function is


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{h(t)}}} = {{{16sin((pi/30)t+3pi/2)+18}}},


with the shift of &nbsp;{{{3pi/2}}}.



This shift converts &nbsp;{{{16sin((pi/30)t+3pi/2)+18}}} &nbsp;&nbsp;into &nbsp;&nbsp;{{{-16cos((pi/30)t)+18}}}.