Trigonometry Examples

$6 - (- 5 + 4 i) - (- 13 - i)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$6 - - 5 - (4 i) - (- 13 - i)$

Step 1.2

Multiply $- 1$ by $- 5$ .

$6 + 5 - (4 i) - (- 13 - i)$

Step 1.3

Multiply $4$ by $- 1$ .

$6 + 5 - 4 i - (- 13 - i)$

Step 1.4

Apply the distributive property.

$6 + 5 - 4 i - - 13 - - i$

Step 1.5

Multiply $- 1$ by $- 13$ .

$6 + 5 - 4 i + 13 - - i$

Step 1.6

Multiply $- 1$ by $- 1$ .

$6 + 5 - 4 i + 13 + 1 i$

Step 1.7

Multiply $i$ by $1$ .

$6 + 5 - 4 i + 13 + i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $6$ and $5$ .

$11 - 4 i + 13 + i$

Step 2.2

Add $11$ and $13$ .

$24 - 4 i + i$

Step 2.3

Add $- 4 i$ and $i$ .

$24 - 3 i$

Step 3

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 4

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 5

Substitute the actual values of $a = 24$ and $b = - 3$ .

$| z | = \sqrt{{(- 3)}^{2} + 24^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

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

$| z | = \sqrt{9 + 24^{2}}$

Step 6.2

Raise $24$ to the power of $2$ .

$| z | = \sqrt{9 + 576}$

Step 6.3

Add $9$ and $576$ .

$| z | = \sqrt{585}$

Step 6.4

Rewrite $585$ as $3^{2} \cdot 65$ .

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Step 6.4.1

Factor $9$ out of $585$ .

$| z | = \sqrt{9 (65)}$

Step 6.4.2

Rewrite $9$ as $3^{2}$ .

$| z | = \sqrt{3^{2} \cdot 65}$

Step 6.5

Pull terms out from under the radical.

$| z | = 3 \sqrt{65}$

Step 7

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

$θ = \arctan (\frac{- 3}{24})$

Step 8

Since inverse tangent of $\frac{- 3}{24}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.12435499$ .

$θ = - 0.12435499$

Step 9

Substitute the values of $θ = - 0.12435499$ and $| z | = 3 \sqrt{65}$ .

$3 \sqrt{65} (\cos (- 0.12435499) + i \sin (- 0.12435499))$