Find All Complex Number Solutions z=5-3i

$z = 5 - 3 i$

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 $a = 5$ and $b = - 3$ .

$| z | = \sqrt{{(- 3)}^{2} + 5^{2}}$

Step 4

Find $| z |$ .

Raise $- 3$ to the power of $2$ .

$| z | = \sqrt{9 + 5^{2}}$

Raise $5$ to the power of $2$ .

$| z | = \sqrt{9 + 25}$

Add $9$ and $25$ .

$| z | = \sqrt{34}$

Step 5

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

$θ = \arctan (\frac{- 3}{5})$

Step 6

Since inverse tangent of $\frac{- 3}{5}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.5404195$ .

$θ = - 0.5404195$

Step 7

Substitute the values of $θ = - 0.5404195$ and $| z | = \sqrt{34}$ .

$\sqrt{34} (\cos (- 0.5404195) + i \sin (- 0.5404195))$

Step 8

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

$z = \sqrt{34} (\cos (- 0.5404195) + i \sin (- 0.5404195))$