Find All Complex Number Solutions z=-302+19.3i

$z = - 302 + 19.3 i$

Step 1

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 2

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 3

Substitute the actual values of $a = - 302$ and $b = 19.3$ .

$| z | = \sqrt{{19.3}^{2} + {(- 302)}^{2}}$

Step 4

Find $| z |$ .

Raise $19.3$ to the power of $2$ .

$| z | = \sqrt{372.49 + {(- 302)}^{2}}$

Raise $- 302$ to the power of $2$ .

$| z | = \sqrt{372.49 + 91204}$

Add $372.49$ and $91204$ .

$| z | = \sqrt{91576.49}$

Step 5

Evaluate the root.

$| z | = 302.6160769$

Step 6

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{19.3}{- 302})$

Step 7

Since inverse tangent of $\frac{19.3}{- 302}$ produces an angle in the second quadrant, the value of the angle is $3.07777215$ .

$θ = 3.07777215$

Step 8

Substitute the values of $θ = 3.07777215$ and $| z | = 302.6160769$ .

$302.6160769 (\cos (3.07777215) + i \sin (3.07777215))$

Step 9

Replace the right side of the equation with the trigonometric form.

$z = 302.6160769 (\cos (3.07777215) + i \sin (3.07777215))$