Trigonometry Examples

2 + 2 i

$2 + 2 i$

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 = 2

$b = 2$ .

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

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

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Raise

2

$2$ to the power of

2

$2$ .

| z | = \sqrt{4 + 2^{2}}

$| z | = \sqrt{4 + 2^{2}}$

Step 4.2

Raise

2

$2$ to the power of

2

$2$ .

| z | = \sqrt{4 + 4}

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

Step 4.3

Add

4

$4$ and

4

$4$ .

| z | = \sqrt{8}

$| z | = \sqrt{8}$

Step 4.4

Rewrite

8

$8$ as

2^{2} \cdot 2

$2^{2} \cdot 2$ .

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Step 4.4.1

Factor

4

$4$ out of

8

$8$ .

| z | = \sqrt{4 (2)}

$| z | = \sqrt{4 (2)}$

Step 4.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 4.5

Pull terms out from under the radical.

| z | = 2 \sqrt{2}

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

| z | = 2 \sqrt{2}

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

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{2}{2})

$θ = \arctan (\frac{2}{2})$

Step 6

Since inverse tangent of

\frac{2}{2}

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

\frac{π}{4}

$\frac{π}{4}$ .

θ = \frac{π}{4}

$θ = \frac{π}{4}$

Step 7

Substitute the values of

θ = \frac{π}{4}

$θ = \frac{π}{4}$ and

| z | = 2 \sqrt{2}

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

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

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