Calculus Examples

$47 + 16 i$

Step 1

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 2

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 3

Substitute the actual values of $a = 47$ and $b = 16$ .

$| z | = \sqrt{16^{2} + 47^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Raise $16$ to the power of $2$ .

$| z | = \sqrt{256 + 47^{2}}$

Step 4.2

Raise $47$ to the power of $2$ .

$| z | = \sqrt{256 + 2209}$

Step 4.3

Add $256$ and $2209$ .

$| z | = \sqrt{2465}$

Step 5

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

$θ = \arctan (\frac{16}{47})$

Step 6

Since inverse tangent of $\frac{16}{47}$ produces an angle in the first quadrant, the value of the angle is $0.32811989$ .

$θ = 0.32811989$

Step 7

Substitute the values of $θ = 0.32811989$ and $| z | = \sqrt{2465}$ .

$\sqrt{2465} (\cos (0.32811989) + i \sin (0.32811989))$