Find All Complex Number Solutions z=-1+ square root of 3i

$z = - 1 + \sqrt{3} i$

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 $a = - 1$ and $b = \sqrt{3}$ .

$| z | = \sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}}$

Step 4

Find $| z |$ .

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \sqrt{{(3^{\frac{1}{2}})}^{2} + {(- 1)}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{3^{\frac{1}{2} \cdot 2} + {(- 1)}^{2}}$

Combine $\frac{1}{2}$ and $2$ .

$| z | = \sqrt{3^{\frac{2}{2}} + {(- 1)}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$| z | = \sqrt{3^{\frac{2}{2}} + {(- 1)}^{2}}$

Rewrite the expression.

$| z | = \sqrt{3 + {(- 1)}^{2}}$

Evaluate the exponent.

$| z | = \sqrt{3 + {(- 1)}^{2}}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{3 + 1}$

Add $3$ and $1$ .

$| z | = \sqrt{4}$

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

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

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

$| z | = 2$

Step 5

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

$θ = \arctan (\frac{\sqrt{3}}{- 1})$

Step 6

Since inverse tangent of $\frac{\sqrt{3}}{- 1}$ produces an angle in the second quadrant, the value of the angle is $\frac{2 π}{3}$ .

$θ = \frac{2 π}{3}$

Step 7

Substitute the values of $θ = \frac{2 π}{3}$ and $| z | = 2$ .

$2 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

Step 8

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

$z = 2 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

Find All Complex Number Solutions z=-1+ square root of 3i

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