Question

Write the complex number in polar form. express the argument in degrees, rounded to the nearest tenth, if necessary. 5) 2 +2i

Answer 1

Here you need to calculate the magnitude and phase angle of the quantity 2 + 2i.

The magnitude is found using the Pyth. Thm. and is sqrt(2^2 + 2^2), or
`√(8)`, or 2
`√(2)`

Recognize that the phase angle is 45 deg, or
`\pi`/4 rad.

Thus, the complex form is (2sqrt(2) angle
`\pi`/4 rad ).

Answer 2

`z=√(8) (cos(45\°) + isin(45\°))`

Explanation:

The polar form of a complex number is

`z=r(cos\theta + isin\theta)`

Where
`r=\sqrt{a^(2) +b^(2) }`,
`a=rcos\theta`,
`b=rsin\theta` and
`\theta = tan^(-1)((b)/(a) )`

In this case, we have the number
`2+2i`, where
`a=2` and
`b=2`, so the module
`r` is

`r=\sqrt{a^(2) +b^(2) }=\sqrt{2^(2) +2^(2) }=√(4+4)=√(8)`

And the angle is

`\theta = tan^(-1)((2)/(2) )\\\theta=tan^(-1)(1)\\\theta = 45\°`

Therefore, the polar form of the given complex number is

`z=√(8) (cos(45\°) + isin(45\°))`