Trigonometry Examples

$5 - 5 \sqrt{3 i}$

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 $b = - 5$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

$| z | = \sqrt{25 + 5^{2}}$

Step 4.2

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

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

Step 4.3

Add $25$ and $25$ .

$| z | = \sqrt{50}$

Step 4.4

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 4.4.1

Factor $25$ out of $50$ .

$| z | = \sqrt{25 (2)}$

Step 4.4.2

Rewrite $25$ as $5^{2}$ .

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

Step 4.5

Pull terms out from under the radical.

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

Step 5

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

$θ = \arctan (\frac{- 5}{5})$

Step 6

Since inverse tangent of $\frac{- 5}{5}$ produces an angle in the fourth 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 $| z | = 5 \sqrt{2}$ .

$5 \sqrt{2} (\cos (- \frac{π}{4}) + i \sin (- \frac{π}{4}))$