Trigonometry Examples

$2 + 2 i$

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 $a = 2$ and $b = 2$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

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

$| z | = \sqrt{4 + 4}$

Step 4.3

Add $4$ and $4$ .

$| z | = \sqrt{8}$

Step 4.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.4.1

Factor $4$ out of $8$ .

$| z | = \sqrt{4 (2)}$

Step 4.4.2

Rewrite $4$ as $2^{2}$ .

$| z | = \sqrt{2^{2} \cdot 2}$

Step 4.5

Pull terms out from under the radical.

$| z | = 2 \sqrt{2}$

Step 5

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

$θ = \arctan (\frac{2}{2})$

Step 6

Since inverse tangent of $\frac{2}{2}$ produces an angle in the first quadrant, the value of the angle is $\frac{π}{4}$ .

$θ = \frac{π}{4}$

Step 7

Substitute the values of $θ = \frac{π}{4}$ and $| z | = 2 \sqrt{2}$ .

$2 \sqrt{2} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$