Convert to Trigonometric Form

$5 i + 3$

Step 1

Reorder $5 i$ and $3$ .

$3 + 5 i$

Step 2

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 3

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 4

Substitute the actual values of $a = 3$ and $b = 5$ .

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

Step 5

Find $| z |$ .

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Step 5.1

Raise $5$ to the power of $2$ .

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

Step 5.2

Raise $3$ to the power of $2$ .

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

Step 5.3

Add $25$ and $9$ .

$| z | = \sqrt{34}$

Step 6

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{5}{3})$

Step 7

Since inverse tangent of $\frac{5}{3}$ produces an angle in the first quadrant, the value of the angle is $1.03037682$ .

$θ = 1.03037682$

Step 8

Substitute the values of $θ = 1.03037682$ and $| z | = \sqrt{34}$ .

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

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Convert to Trigonometric Form

Complex Numbers

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