Question 870024
{{{y=1+4cos(2t)-4sin(4t)=1+4}}}{{{"["}}}{{{cos(2t)-sin(4t)}}}{{{"]"}}} is a periodic function, waving above and below {{{y=1}}} , whose graph looks like this:
{{{graph(600,300,-1.6,6.4,-7,9,1+4cos(2x)-4sin(4x),1)}}} Its period is {{{pi}}} because at {{{t}}} and {{{t+pi}}} the function has the same {{{y}}} value:
{{{y=1+4cos(2t+2pi)-4sin(4t+4pi)=y=1+4cos(2t)-4sin(4t)}}} .
The function will take the value {{{y=1}}} an infinite number of times.
For an answer to part a) we can list the first few in order and then write ", ..." or we can write it as a formula.
Either way, it is not a simple answer.
 
The double angle trigonometric identity
{{{sin(2A)=2sin(A)*cos(A)}}} can be applied, with {{{A=2t}}} to get
{{{y=1+4}}}{{{"["}}}{{{cos(2t)-2sin(2t)cos(2t)}}}{{{"]"}}}
Taking out {{{cos(2t)}}} as a common factor, we get
{{{y=1+4}}}{{{"["}}}{{{cos(2t)(1-2sin(2t))}}}{{{"]"}}}
{{{y=1+4*cos(2t)*(1-2sin(2t))}}}
When will we have {{{y=1}}} ?
We have to solve
{{{1=1+4*cos(2t)*(1-2sin(2t))}}}<--->{{{4*cos(2t)*(1-2sin(2t))=0}}}
That will be true when one of those factors (either {{{cos(2t)}}} or {{{(1-2sin(2t))}}} ) is zero.
 
{{{cos(2t)=0}}} --> {{{2t=pi/2}}} or {{{2t=3pi/2}}} or anything you can get by adding {{{2pi}}} to a previous answer.
So {{{t=pi/4}}} or {{{t=3pi/4}}} or anything you can get by adding {{{pi}}} to a previous answer.
 
{{{1-2sin(2t)=0}}}<--->{{{sin(2t)=1/2}}}<--->{{{2t=pi/6}}} or {{{2t=5pi/6}}} or anything you can get by adding {{{2pi}}} to a previous answer.
That means {{{t=pi/12}}} or {{{t=5pi/12}}} or anything you can get by adding {{{pi}}} to a previous answer.
 
a) The list of times when {{{y=0}}} is
{{{pi/12}}}{{{","}}} {{{pi/4=3pi/12}}}{{{","}}} {{{5pi/12}}}{{{","}}} {{{3pi/4=9pi/12}}}{{{","}}} {{{13pi/12=pi/12+pi}}}{{{","}}}{{{"..."}}} (and keep adding {{{pi}}} to previous answers).
You could state that the answers are
{{{t=(n+1)pi/12}}} , {{{t=(n+3)pi/12}}} , {{{t=(n+5)pi/12}}} , and {{{t=(n+9)pi/12}}} , for any non-negative integer {{{n}}} .
I could not think of a simple and elegant way to express that as one formula.
 
b) The first time {{{y=1}}} happens when {{{highlight(t=pi/12=0.26)}}}