Find All Complex Number Solutions z=3-4i

$z = 3 - 4 i$

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

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

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

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

Substitute the actual values of $a = 3$ and $b = - 4$ .

$| z | = \sqrt{{(- 4)}^{2} + 3^{2}}$

Find $| z |$ .

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Raise $- 4$ to the power of $2$ .

$| z | = \sqrt{16 + 3^{2}}$

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

$| z | = \sqrt{16 + 9}$

Add $16$ and $9$ .

$| z | = \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 5$

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

$θ = \arctan (\frac{- 4}{3})$

Since inverse tangent of $\frac{- 4}{3}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.92729521$ .

$θ = - 0.92729521$

Substitute the values of $θ = - 0.92729521$ and $| z | = 5$ .

$5 (\cos (- 0.92729521) + i \sin (- 0.92729521))$

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

$z = 5 (\cos (- 0.92729521) + i \sin (- 0.92729521))$

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