# Express the complex number -243 in polar form, giving the principal value of the argumentI think the answer is <243, pi> that is the modulus is r= 243 and argument = pi but how can I show this

You need to convert the given form of complex number in polar form, hence you need to find r and the argument such that:

`|r| = sqrt((-243)^2 + 0^2)`

The distance `|r|`  is hypotenuse in right triangle that has the lengths of the legs -243 and 0, hence, you may use Pythagorean theorem...

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You need to convert the given form of complex number in polar form, hence you need to find r and the argument such that:

`|r| = sqrt((-243)^2 + 0^2)`

The distance `|r|`  is hypotenuse in right triangle that has the lengths of the legs -243 and 0, hence, you may use Pythagorean theorem to find r.

`|r| = 243`

You need to write the polar form such that:

z = |r|*(cos alpha + i sin alpha)

You need to find the argument `alpha`  such that:

`tan alpha = b/a =gt tan alpha = 0/(-243)`

`tan alpha = pi, `

Hence, evaluating the argument `alpha`    yields `pi` .

You need to substitute `pi ` for `alpha ` and 243 for |r| in polar form such that:

`z = 243(cos pi + i sin pi)`

Hence, converting the complex number to polar form yields `z = 243(cos pi + i sin pi).`

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