# 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

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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).`

johnsmith3000 | Student, Undergraduate | eNotes Newbie

Posted on

just to note i think the guy above however smart he seems has given the answer in cartesian form not polar,i aint no maths wizz but hey hope one of us has helped u

johnsmith3000 | Student, Undergraduate | eNotes Newbie

Posted on

you do not need to show it, the <243,pi> is giving it in polar form in the principle arguement, as the principle arguement is when theta is larger than negative pi,and less than or equal to pi

hope this helps,had same problem till everything clicked into place