Reformatted question: Find exact value of #cos(-(11pi)/2)#
First, we can find a coterminal angle in the interval #(-2pi,2pi)# by finding the remainder when #(11pi)/2# is divided by #2pi# which is the angle in radians of one circle (disregard the negative sign for now, we'll add it back on after we divide):
#(5.5pi)/(2pi)=(2)(2pi)+1.5pi#
Therefore the remainder is #1.5pi# or #(3pi)/2#.
Now, we know a coterminal angle of #(-11pi)/2# is #-(3pi)/2# (here we add the negative sign back):
#cos(-(3pi)/2)#
We can write #cos(-(3pi)/2)# as #cos(pi/2)# since #-(3pi)/2# and #pi/2# are also coterminal because #-(3pi)/2# is #3/4# around the unit circle going clockwise while #pi/2# is #1/4# around the unit circle going counterclockwise:
#thereforecos(-(3pi)/2)=cos(pi/2)=0# since it is one of our special unit circle trig values
