#" "#
Given the Complex Number: #color(blue)(-2i#
The standard form of a complex number is #color(red)(z = a+bi#
So, we have #z=-2i = 0-2i# with #a=0 and b=-2#
The complex number #color(blue)(-2i# is marked on a Complex Plane
Observe that #color(blue)(-2i# lies on the imaginary axis and it takes the same position as #A=(0,-2)# in Quadrant-3.
This point makes a #color(red)(270^@)# from the Real Axis measured in the counter-clockwise direction.
#color(red)(270^@)=(3pi)/2# Radians.
Using the formula,
#color(red)(z=r(cos theta + i sin theta)#
#r# is the Modulus of z, #color(red)(|z|=|a+bi|=sqrt(a^2+b^2#
#theta# is the argument.
we can represent the complex number #Z=0-2i# in trigonometric form as follows:
#r# is the radius, which is #2# in this problem.
Hence,
#z=2[cos((3pi)/2)+i sin ((3pi)/2)]#
Hope it helps.
