Trigonometry Examples

$4 i - 2$

Step 1

Reorder

$4 i$ and

$- 2$ .

$- 2 + 4 i$

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

$a = - 2$ and

$b = 4$ .

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

Step 5

Find

$| z |$ .

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Step 5.1

Raise

$4$ to the power of

$2$ .

$| z | = \sqrt{16 + {(- 2)}^{2}}$

Step 5.2

Raise

$- 2$ to the power of

$2$ .

$| z | = \sqrt{16 + 4}$

Step 5.3

Add

$16$ and

$4$ .

$| z | = \sqrt{20}$

Step 5.4

Rewrite

$20$ as

$2^{2} \cdot 5$ .

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Step 5.4.1

Factor

$4$ out of

$20$ .

$| z | = \sqrt{4 (5)}$

Step 5.4.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 5.5

Pull terms out from under the radical.

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

Step 6

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

$θ = \arctan (\frac{4}{- 2})$

Step 7

Since inverse tangent of

$\frac{4}{- 2}$ produces an angle in the second quadrant, the value of the angle is

$2.03444393$ .

$θ = 2.03444393$

Step 8

Substitute the values of

$θ = 2.03444393$ and

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

$2 \sqrt{5} (\cos (2.03444393) + i \sin (2.03444393))$

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