Trigonometry Examples

$3 i - 2$

Step 1

Reorder

$3 i$ and

$- 2$ .

$- 2 + 3 i$

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

$b = 3$ .

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

Step 5

Find

$| z |$ .

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Step 5.1

Raise

$3$ to the power of

$2$ .

$| z | = \sqrt{9 + {(- 2)}^{2}}$

Step 5.2

Raise

$- 2$ to the power of

$2$ .

$| z | = \sqrt{9 + 4}$

Step 5.3

Add

$9$ and

$4$ .

$| z | = \sqrt{13}$

Step 6

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

$θ = \arctan (\frac{3}{- 2})$

Step 7

Since inverse tangent of

$\frac{3}{- 2}$ produces an angle in the second quadrant, the value of the angle is

$2.15879893$ .

$θ = 2.15879893$

Step 8

Substitute the values of

$θ = 2.15879893$ and

$| z | = \sqrt{13}$ .

$\sqrt{13} (\cos (2.15879893) + i \sin (2.15879893))$

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