Question 1130301
 {{{y = 3csc(x/2)}}} Show the asymptotes and the extrema.

{{{y = 3(1/sin(x/2))}}} 


{{{ graph( 600, 600, -30, 30, -10, 10,3(1/sin(x/2))) }}}


vertical asymptote {{{x=2pi*n}}}  where {{{n}}} is an integer

{{{y = 3csc(x/2)}}} is asymptotic to {{{x = 0}}}
{{{y = 3 csc(x/2) }}}is asymptotic to {{{x = 2pi (2n + 1)}}}
{{{y = 3 csc(x/2) }}}is asymptotic to {{{x = 4pi*n}}}

asymptote: 

if {{{n=0}}},  {{{x = 2pi (2*0 + 1)}}}-> {{{x = 2pi }}}
if {{{n=0}}}, {{{x = 4pi*0}}}->{{{x=0}}}

if {{{n=1}}},  {{{x = 2pi (2*1 + 1)}}}-> {{{x = 6pi }}}
if {{{n=1}}}, {{{x = 4pi*1}}}->{{{x=4pi}}}

and so on


{{{drawing ( 600, 600, -30, 30, -10, 10,
line(4pi,10,4pi,-10),line(2pi,10,2pi,-10),green(line(0,10,0,-10)),
line(8pi,10,8pi,-10),line(6pi,10,6pi,-10),line(-6pi,10,-6pi,-10),
line(-4pi,10,-4pi,-10),line(-2pi,10,-2pi,-10),line(-8pi,10,-8pi,-10),
graph( 600, 600, -30, 30, -10, 10,3(1/sin(x/2)))) }}}



period {{{2pi/b}}} ,and {{{b=1/2}}}->{{{2pi/(1/2)=4pi}}}

amplitude: none

max{{{3csc(x/2) = -3}}} at {{{x = 4pi* n - pi}}} for integer{{{ n}}}

min{{{3csc(x/2) = 3}}} at {{{x = 4pi*n  + pi}}} for integer {{{n}}}