21) find the exact value without a calculator. csc 4pi/3?

Question

please show your work briefly, thank you

Answer 1

csc(4pi/3) = 1 / sin(4pi/3)

and sin(4pi/3) = - sqrt(3) / 2

so 1 / sin(4pi/3) = 1 / (-sqrt(3) / 2) = -2 / sqrt(3)

csc(4pi/3) = -2 / sqrt(3) = -2 * sqrt(3) / 3

Answer 2

csc ( 4π / 3 ) = 1 / sin ( 4π / 3 )

sin ( 4π / 3 ) = - sin ( 4π / 3 - π ) = - sin ( π / 3 ) = - sqrt ( 3 ) / 2

So csc ( 4π / 3 ) = - 2 / sqrt ( 3 )

Answer 3

csc(4pi/3)=1/sin(4pi/3)
4pi/3=pi+pi/3
sin(pi+pi/3)=sin(pi)cos(pi/3)+cos(pi)sin(pi/3)
sin(pi)=0
cos)pi)=-1
sin(pi/3)=√3/2
so csc(4pi/3)=-2/√3