Trigonometry Examples

$z = - 5 + 3 i$

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 $a = - 5$ and $b = 3$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

$| z | = \sqrt{9 + {(- 5)}^{2}}$

Step 4.2

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

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

Step 4.3

Add $9$ and $25$ .

$| z | = \sqrt{34}$

Step 5

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

$θ = \arctan (\frac{3}{- 5})$

Step 6

Since inverse tangent of $\frac{3}{- 5}$ produces an angle in the second quadrant, the value of the angle is $2.60117315$ .

$θ = 2.60117315$

Step 7

Substitute the values of $θ = 2.60117315$ and $| z | = \sqrt{34}$ .

$\sqrt{34} (\cos (2.60117315) + i \sin (2.60117315))$

Step 8

Replace the right side of the equation with the trigonometric form.

$z = \sqrt{34} (\cos (2.60117315) + i \sin (2.60117315))$