Trigonometry Examples

$4 + 9 i$

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 $a = 4$ and $b = 9$ .

$| z | = \sqrt{9^{2} + 4^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Raise $9$ to the power of $2$ .

$| z | = \sqrt{81 + 4^{2}}$

Step 4.2

Raise $4$ to the power of $2$ .

$| z | = \sqrt{81 + 16}$

Step 4.3

Add $81$ and $16$ .

$| z | = \sqrt{97}$

Step 5

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

$θ = \arctan (\frac{9}{4})$

Step 6

Since inverse tangent of $\frac{9}{4}$ produces an angle in the first quadrant, the value of the angle is $1.15257199$ .

$θ = 1.15257199$

Step 7

Substitute the values of $θ = 1.15257199$ and $| z | = \sqrt{97}$ .

$\sqrt{97} (\cos (1.15257199) + i \sin (1.15257199))$