Find All Complex Number Solutions z=8i-172

$z = 8 i - 172$

Step 1

Reorder $8 i$ and $- 172$ .

$z = - 172 + 8 i$

Step 2

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 3

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 4

Substitute the actual values of $a = - 172$ and $b = 8$ .

$| z | = \sqrt{8^{2} + {(- 172)}^{2}}$

Step 5

Find $| z |$ .

Raise $8$ to the power of $2$ .

$| z | = \sqrt{64 + {(- 172)}^{2}}$

Raise $- 172$ to the power of $2$ .

$| z | = \sqrt{64 + 29584}$

Add $64$ and $29584$ .

$| z | = \sqrt{29648}$

Rewrite $29648$ as $4^{2} \cdot 1853$ .

Factor $16$ out of $29648$ .

$| z | = \sqrt{16 (1853)}$

Rewrite $16$ as $4^{2}$ .

$| z | = \sqrt{4^{2} \cdot 1853}$

Pull terms out from under the radical.

$| z | = 4 \sqrt{1853}$

Step 6

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

$θ = \arctan (\frac{8}{- 172})$

Step 7

Since inverse tangent of $\frac{8}{- 172}$ produces an angle in the second quadrant, the value of the angle is $3.09511452$ .

$θ = 3.09511452$

Step 8

Substitute the values of $θ = 3.09511452$ and $| z | = 4 \sqrt{1853}$ .

$4 \sqrt{1853} (\cos (3.09511452) + i \sin (3.09511452))$

Step 9

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

$z = 4 \sqrt{1853} (\cos (3.09511452) + i \sin (3.09511452))$