Find All Complex Number Solutions z=9+3i

$z = 9 + 3 i$

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 $a = 9$ and $b = 3$ .

$| z | = \sqrt{3^{2} + 9^{2}}$

Step 4

Find $| z |$ .

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

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

Raise $9$ to the power of $2$ .

$| z | = \sqrt{9 + 81}$

Add $9$ and $81$ .

$| z | = \sqrt{90}$

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$| z | = \sqrt{9 (10)}$

Rewrite $9$ as $3^{2}$ .

$| z | = \sqrt{3^{2} \cdot 10}$

Pull terms out from under the radical.

$| z | = 3 \sqrt{10}$

Step 5

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

$θ = \arctan (\frac{3}{9})$

Step 6

Since inverse tangent of $\frac{3}{9}$ produces an angle in the first quadrant, the value of the angle is $0.32175055$ .

$θ = 0.32175055$

Step 7

Substitute the values of $θ = 0.32175055$ and $| z | = 3 \sqrt{10}$ .

$3 \sqrt{10} (\cos (0.32175055) + i \sin (0.32175055))$

Step 8

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

$z = 3 \sqrt{10} (\cos (0.32175055) + i \sin (0.32175055))$