Question

How do you write the complex number in trigonometric form `-2`?

Answer 1

`2(cos(pi)+isin(pi))`

Explanation:

`2=2+i * 0 or color(red)2(color(blue)1+i * color(blue)0)`

The general trigonometric for a complex number is
`color(white)("XXX")color(red)r(cos(color(blue)pi)+i * sin(color(blue)pi))`

So we need a value `color(blue)theta`
such that
`color(white)("XXX")cos(color(blue)theta)=1color(white)("XX")andcolor(white)("XX")sin(color(blue)theta)=color(blue)0`

The simplest solution that meets these requirements is `color(blue)theta =color(blue)pi`

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To be complete, we should possibly note that any
`color(white)("XXX")color(blue)theta = color(blue)pi + k2pi, k in ZZ`
would also meet the stated requirement
and the "complete" answer would be
`color(white)("XXX")2(cos(pi+k2pi)+isin(pi+k2pi))), k in ZZ`

While this is correct, I personally find that it obscures the significant components of the answer.