Trigonometry Examples

$4 i - 2$

Step 1

Reorder $4 i$ and $- 2$ .

$- 2 + 4 i$

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 $a = - 2$ and $b = 4$ .

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

Step 5

Find $| z |$ .

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Step 5.1

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

$| z | = \sqrt{16 + {(- 2)}^{2}}$

Step 5.2

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

$| z | = \sqrt{16 + 4}$

Step 5.3

Add $16$ and $4$ .

$| z | = \sqrt{20}$

Step 5.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 5.4.1

Factor $4$ out of $20$ .

$| z | = \sqrt{4 (5)}$

Step 5.4.2

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

$| z | = \sqrt{2^{2} \cdot 5}$

Step 5.5

Pull terms out from under the radical.

$| z | = 2 \sqrt{5}$

Step 6

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

$θ = \arctan (\frac{4}{- 2})$

Step 7

Since inverse tangent of $\frac{4}{- 2}$ produces an angle in the second quadrant, the value of the angle is $2.03444393$ .

$θ = 2.03444393$

Step 8

Substitute the values of $θ = 2.03444393$ and $| z | = 2 \sqrt{5}$ .

$2 \sqrt{5} (\cos (2.03444393) + i \sin (2.03444393))$

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