Can you help me understand how to find the trigonometric form of -4?

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The trigonometric form of a complex number is

`r(cos(phi)+i*sin(phi)),`

where `r` is a positive number and `phi` is a number in` [0, 2pi).`

Here `r` is the absolute value of a complex number and `phi` is the argument. Any nonzero complex number has one and only one such representation.

If we plot a complex number on a coordinate plane and draw a directed segment from the origin to the number's point, then `r` will be the length of this segment and `phi` will be the angle between the positive half of the x-axis and the segment.

For the number `-4=-4+0i` the absolute value is obviously `4=sqrt((-4)^2+0^2).`

What about `phi`? The point `(-4, 0)` lies at the negative half of the x-axis and the angle from the positive half to the negative is obviously `pi` (a half of an entire circumference, which is `2pi`).

We can check this: `4(cos(pi)+i*sin(pi))=4*(-1+0*i)=-4` (true).

So the answer is `-4=4(cos(pi)+i*sin(pi)).`

 

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