How do you write the trigonometric form of #-2i#?

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Answer 1 · 2018-04-17T06:10:15

Please read the explanation.

#" "#
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

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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.

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#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.