Question 1205131
 a rope {{{3m }}}long
 a ceiling {{{4m}}} high


so, from the floor to the pendulum is {{{1m }}}=> minimum height 

the vertical distance to the ceiling is :

{{{h=3m*cos(alpha) }}}

since the highest angle is just {{{alpha= pi/3 }}}and {{{cos(pi/3) = 1/2}}}, the smallest distance to the ceiling is {{{h=3m*(1/2) = 1.5m }}}
 
and 
the highest point is {{{h[1]=1m+1.5m=2.5m}}} from the ground 


then, cosine function will be:

{{{h(t)=D-Acos(Bt)}}}

range of the pendulum's height above the ground is 

{{{1<= h(t) <= 2.5}}}, where 

{{{h(0)=1}}}
{{{h(2)=2.5}}}
{{{h(4)=1}}}

amplitude is {{{A=(2.5-1)/2=0.75}}}

midline is {{{D=(1+2.5)/2=1.75}}}

the period is {{{T=4 }}}=>{{{B=2pi/T=2pi/4=pi/2}}}

and, function is:

{{{h(t)=1.75-0.75*cos((pi/2)*t)}}}

​

{{{drawing( 600, 600, -6, 6, -2, 5,
circle(0,1,.15), circle(2,2.5,.15),green(line(0.08,4,0.08,1)),locate(0.3,2.5,3m),
green(line(2,2.5,0.08,4)),locate(0.3,0.5,1m),blue(line(0.08,0,0.08,1)),locate(2,1.5,2.5m),blue(line(2,0,2,2.5)),
 graph( 600, 600, -6, 6, -2, 5, 4,4,1.75-0.75*cos((pi/2)*x))) }}}