Linear Algebra Examples

$- 6 - 2 i$

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 $a = - 6$ and $b = - 2$ .

$| z | = \sqrt{{(- 2)}^{2} + {(- 6)}^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

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

$| z | = \sqrt{4 + {(- 6)}^{2}}$

Step 4.2

Raise $- 6$ to the power of $2$ .

$| z | = \sqrt{4 + 36}$

Step 4.3

Add $4$ and $36$ .

$| z | = \sqrt{40}$

Step 4.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 4.4.1

Factor $4$ out of $40$ .

$| z | = \sqrt{4 (10)}$

Step 4.4.2

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

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

Step 4.5

Pull terms out from under the radical.

$| z | = 2 \sqrt{10}$

Step 5

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

$θ = \arctan (\frac{- 2}{- 6})$

Step 6

Since inverse tangent of $\frac{- 2}{- 6}$ produces an angle in the third quadrant, the value of the angle is $3.4633432$ .

$θ = 3.4633432$

Step 7

Substitute the values of $θ = 3.4633432$ and $| z | = 2 \sqrt{10}$ .

$2 \sqrt{10} (\cos (3.4633432) + i \sin (3.4633432))$