Trigonometry Examples

2 + 3 i

$2 + 3 i$

Step 1

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

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

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

Step 2

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

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

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

z = a + b i

$z = a + b i$

Step 3

Substitute the actual values of

a = 2

$a = 2$ and

b = 3

$b = 3$ .

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

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

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Raise

3

$3$ to the power of

2

$2$ .

| z | = \sqrt{9 + 2^{2}}

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

Step 4.2

Raise

2

$2$ to the power of

2

$2$ .

| z | = \sqrt{9 + 4}

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

Step 4.3

Add

9

$9$ and

4

$4$ .

| z | = \sqrt{13}

$| z | = \sqrt{13}$

| z | = \sqrt{13}

$| z | = \sqrt{13}$

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{3}{2})

$θ = \arctan (\frac{3}{2})$

Step 6

Since inverse tangent of

\frac{3}{2}

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

0.98279372

$0.98279372$ .

θ = 0.98279372

$θ = 0.98279372$

Step 7

Substitute the values of

θ = 0.98279372

$θ = 0.98279372$ and

| z | = \sqrt{13}

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

\sqrt{13} (cos (0.98279372) + i sin (0.98279372))

$\sqrt{13} (\cos (0.98279372) + i \sin (0.98279372))$