Question 1115325
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A cosine function shifted one-quarter period to the right is the same as a sine function:<br>
{{{cos(t-pi/2) = cos(t)*cos(pi/2)+sin(t)*sin(pi/2) = cos(t)*0+sin(t)*1 = sin(t)}}}<br>
So the given function<br>
{{{r(t) = -4sin(5t+pi/2)+7}}}<br>
is the same as the function<br>
{{{v(t) = -4cos((5t+pi/2)-pi/2)+7}}}<br>
or<br>
{{{v(t) = -4cos(5t)+7}}}<br>
There are of course an infinite number of equivalent functions using either sine or cosine.<br>
For example, a cosine function shifted a quarter period to the left is the same as a negative sine function:<br>
{{{cos(t+pi/2) = cos(t)*cos(pi/2)-sin(t)*sin(pi/2) = cos(t)*0+sin(t)*(-1) = -sin(t)}}}<br>
So another cosine function equivalent to the given sine function is<br>
{{{v(t) = 4cos((5t+pi/2)+pi/2)+7}}}<br>
or<br>
{{{v(t) = 4cos(5t+pi)+7}}}<br>