Algebra Examples

z = 9 + 3 i

$z = 9 + 3 i$

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 = 9

$a = 9$ and

b = 3

$b = 3$ .

| z | = \sqrt{3^{2} + 9^{2}}

$| z | = \sqrt{3^{2} + 9^{2}}$

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 4.2

Raise

9

$9$ to the power of

2

$2$ .

| z | = \sqrt{9 + 81}

$| z | = \sqrt{9 + 81}$

Step 4.3

Add

9

$9$ and

81

$81$ .

| z | = \sqrt{90}

$| z | = \sqrt{90}$

Step 4.4

Rewrite

90

$90$ as

3^{2} \cdot 10

$3^{2} \cdot 10$ .

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Step 4.4.1

Factor

9

$9$ out of

90

$90$ .

| z | = \sqrt{9 (10)}

$| z | = \sqrt{9 (10)}$

Step 4.4.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

| z | = \sqrt{3^{2} \cdot 10}

$| z | = \sqrt{3^{2} \cdot 10}$

| z | = \sqrt{3^{2} \cdot 10}

$| z | = \sqrt{3^{2} \cdot 10}$

Step 4.5

Pull terms out from under the radical.

| z | = 3 \sqrt{10}

$| z | = 3 \sqrt{10}$

| z | = 3 \sqrt{10}

$| z | = 3 \sqrt{10}$

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{3}{9})

$θ = \arctan (\frac{3}{9})$

Step 6

Since inverse tangent of

\frac{3}{9}

$\frac{3}{9}$ produces an angle in the first quadrant, the value of the angle is

0.32175055

$0.32175055$ .

θ = 0.32175055

$θ = 0.32175055$

Step 7

Substitute the values of

θ = 0.32175055

$θ = 0.32175055$ and

| z | = 3 \sqrt{10}

$| z | = 3 \sqrt{10}$ .

3 \sqrt{10} (cos (0.32175055) + i sin (0.32175055))

$3 \sqrt{10} (\cos (0.32175055) + i \sin (0.32175055))$

Step 8

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

z = 3 \sqrt{10} (cos (0.32175055) + i sin (0.32175055))

$z = 3 \sqrt{10} (\cos (0.32175055) + i \sin (0.32175055))$