Linear Algebra Examples

- 7 + 7 i

$- 7 + 7 i$

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 = - 7

$a = - 7$ and

b = 7

$b = 7$ .

| z | = \sqrt{7^{2} + {(- 7)}^{2}}

$| z | = \sqrt{7^{2} + {(- 7)}^{2}}$

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Raise

7

$7$ to the power of

2

$2$ .

| z | = \sqrt{49 + {(- 7)}^{2}}

$| z | = \sqrt{49 + {(- 7)}^{2}}$

Step 4.2

Raise

- 7

$- 7$ to the power of

2

$2$ .

| z | = \sqrt{49 + 49}

$| z | = \sqrt{49 + 49}$

Step 4.3

Add

49

$49$ and

49

$49$ .

| z | = \sqrt{98}

$| z | = \sqrt{98}$

Step 4.4

Rewrite

98

$98$ as

7^{2} \cdot 2

$7^{2} \cdot 2$ .

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Step 4.4.1

Factor

49

$49$ out of

98

$98$ .

| z | = \sqrt{49 (2)}

$| z | = \sqrt{49 (2)}$

Step 4.4.2

Rewrite

49

$49$ as

7^{2}

$7^{2}$ .

| z | = \sqrt{7^{2} \cdot 2}

$| z | = \sqrt{7^{2} \cdot 2}$

| z | = \sqrt{7^{2} \cdot 2}

$| z | = \sqrt{7^{2} \cdot 2}$

Step 4.5

Pull terms out from under the radical.

| z | = 7 \sqrt{2}

$| z | = 7 \sqrt{2}$

| z | = 7 \sqrt{2}

$| z | = 7 \sqrt{2}$

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{7}{- 7})

$θ = \arctan (\frac{7}{- 7})$

Step 6

Since inverse tangent of

\frac{7}{- 7}

$\frac{7}{- 7}$ produces an angle in the second quadrant, the value of the angle is

\frac{3 π}{4}

$\frac{3 π}{4}$ .

θ = \frac{3 π}{4}

$θ = \frac{3 π}{4}$

Step 7

Substitute the values of

θ = \frac{3 π}{4}

$θ = \frac{3 π}{4}$ and

| z | = 7 \sqrt{2}

$| z | = 7 \sqrt{2}$ .

7 \sqrt{2} (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4}))

$7 \sqrt{2} (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4}))$