Trigonometry Examples

5 i + 3

$5 i + 3$

Step 1

Reorder

5 i

$5 i$ and

3

$3$ .

3 + 5 i

$3 + 5 i$

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 = 3

$a = 3$ and

b = 5

$b = 5$ .

| z | = \sqrt{5^{2} + 3^{2}}

$| z | = \sqrt{5^{2} + 3^{2}}$

Step 5

Find

| z |

$| z |$ .

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Step 5.1

Raise

5

$5$ to the power of

2

$2$ .

| z | = \sqrt{25 + 3^{2}}

$| z | = \sqrt{25 + 3^{2}}$

Step 5.2

Raise

3

$3$ to the power of

2

$2$ .

| z | = \sqrt{25 + 9}

$| z | = \sqrt{25 + 9}$

Step 5.3

Add

25

$25$ and

9

$9$ .

| z | = \sqrt{34}

$| z | = \sqrt{34}$

| z | = \sqrt{34}

$| z | = \sqrt{34}$

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{5}{3})

$θ = \arctan (\frac{5}{3})$

Step 7

Since inverse tangent of

\frac{5}{3}

$\frac{5}{3}$ produces an angle in the first quadrant, the value of the angle is

1.03037682

$1.03037682$ .

θ = 1.03037682

$θ = 1.03037682$

Step 8

Substitute the values of

θ = 1.03037682

$θ = 1.03037682$ and

| z | = \sqrt{34}

$| z | = \sqrt{34}$ .

\sqrt{34} (\cos (1.03037682) + i \sin (1.03037682))

$\sqrt{34} (\cos (1.03037682) + i \sin (1.03037682))$

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