Linear Algebra Examples

8 i + 6

$8 i + 6$

Step 1

Reorder

8 i

$8 i$ and

6

$6$ .

6 + 8 i

$6 + 8 i$

Step 2

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 3

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 4

Substitute the actual values of

a = 6

$a = 6$ and

b = 8

$b = 8$ .

| z | = \sqrt{8^{2} + 6^{2}}

$| z | = \sqrt{8^{2} + 6^{2}}$

Step 5

Find

| z |

$| z |$ .

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Step 5.1

Raise

8

$8$ to the power of

2

$2$ .

| z | = \sqrt{64 + 6^{2}}

$| z | = \sqrt{64 + 6^{2}}$

Step 5.2

Raise

6

$6$ to the power of

2

$2$ .

| z | = \sqrt{64 + 36}

$| z | = \sqrt{64 + 36}$

Step 5.3

Add

64

$64$ and

36

$36$ .

| z | = \sqrt{100}

$| z | = \sqrt{100}$

Step 5.4

Rewrite

100

$100$ as

10^{2}

$10^{2}$ .

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

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

Step 5.5

Pull terms out from under the radical, assuming positive real numbers.

| z | = 10

$| z | = 10$

| z | = 10

$| z | = 10$

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{8}{6})

$θ = \arctan (\frac{8}{6})$

Step 7

Since inverse tangent of

\frac{8}{6}

$\frac{8}{6}$ produces an angle in the first quadrant, the value of the angle is

0.92729521

$0.92729521$ .

θ = 0.92729521

$θ = 0.92729521$

Step 8

Substitute the values of

θ = 0.92729521

$θ = 0.92729521$ and

| z | = 10

$| z | = 10$ .

10 (\cos (0.92729521) + i \sin (0.92729521))

$10 (\cos (0.92729521) + i \sin (0.92729521))$

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