Precalculus Examples

$\frac{2}{3} + 6 i + \frac{5}{2} \cdot i$

Step 1

Combine $\frac{5}{2}$ and $i$ .

$\frac{2}{3} + 6 i + \frac{5 i}{2}$

Step 2

To write $6 i$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2}{3} + 6 i \cdot \frac{2}{2} + \frac{5 i}{2}$

Step 3

Combine $6 i$ and $\frac{2}{2}$ .

$\frac{2}{3} + \frac{6 i \cdot 2}{2} + \frac{5 i}{2}$

Step 4

Combine the numerators over the common denominator.

$\frac{2}{3} + \frac{17 i}{2}$

Step 5

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 6

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 7

Substitute the actual values of $a = \frac{2}{3}$ and $b = \frac{17}{2}$ .

$| z | = \sqrt{{(\frac{17}{2})}^{2} + {(\frac{2}{3})}^{2}}$

Step 8

Find $| z |$ .

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Step 8.1

Apply the product rule to $\frac{17}{2}$ .

$| z | = \sqrt{\frac{17^{2}}{2^{2}} + {(\frac{2}{3})}^{2}}$

Step 8.2

Raise $17$ to the power of $2$ .

$| z | = \sqrt{\frac{289}{2^{2}} + {(\frac{2}{3})}^{2}}$

Step 8.3

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

$| z | = \sqrt{\frac{289}{4} + {(\frac{2}{3})}^{2}}$

Step 8.4

Apply the product rule to $\frac{2}{3}$ .

$| z | = \sqrt{\frac{289}{4} + \frac{2^{2}}{3^{2}}}$

Step 8.5

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

$| z | = \sqrt{\frac{289}{4} + \frac{4}{3^{2}}}$

Step 8.6

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

$| z | = \sqrt{\frac{289}{4} + \frac{4}{9}}$

Step 8.7

To write $\frac{289}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$| z | = \sqrt{\frac{289}{4} \cdot \frac{9}{9} + \frac{4}{9}}$

Step 8.8

To write $\frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$| z | = \sqrt{\frac{289}{4} \cdot \frac{9}{9} + \frac{4}{9} \cdot \frac{4}{4}}$

Step 8.9

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.9.1

Multiply $\frac{289}{4}$ by $\frac{9}{9}$ .

$| z | = \sqrt{\frac{289 \cdot 9}{4 \cdot 9} + \frac{4}{9} \cdot \frac{4}{4}}$

Step 8.9.2

Multiply $4$ by $9$ .

$| z | = \sqrt{\frac{289 \cdot 9}{36} + \frac{4}{9} \cdot \frac{4}{4}}$

Step 8.9.3

Multiply $\frac{4}{9}$ by $\frac{4}{4}$ .

$| z | = \sqrt{\frac{289 \cdot 9}{36} + \frac{4 \cdot 4}{9 \cdot 4}}$

Step 8.9.4

Multiply $9$ by $4$ .

$| z | = \sqrt{\frac{289 \cdot 9}{36} + \frac{4 \cdot 4}{36}}$

Step 8.10

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{289 \cdot 9 + 4 \cdot 4}{36}}$

Step 8.11

Simplify the numerator.

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Step 8.11.1

Multiply $289$ by $9$ .

$| z | = \sqrt{\frac{2601 + 4 \cdot 4}{36}}$

Step 8.11.2

Multiply $4$ by $4$ .

$| z | = \sqrt{\frac{2601 + 16}{36}}$

Step 8.11.3

Add $2601$ and $16$ .

$| z | = \sqrt{\frac{2617}{36}}$

Step 8.12

Rewrite $\sqrt{\frac{2617}{36}}$ as $\frac{\sqrt{2617}}{\sqrt{36}}$ .

$| z | = \frac{\sqrt{2617}}{\sqrt{36}}$

Step 8.13

Simplify the denominator.

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Step 8.13.1

Rewrite $36$ as $6^{2}$ .

$| z | = \frac{\sqrt{2617}}{\sqrt{6^{2}}}$

Step 8.13.2

Pull terms out from under the radical, assuming positive real numbers.

$| z | = \frac{\sqrt{2617}}{6}$

Step 9

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

$θ = \arctan (\frac{\frac{17}{2}}{\frac{2}{3}})$

Step 10

Since inverse tangent of $\frac{\frac{17}{2}}{\frac{2}{3}}$ produces an angle in the first quadrant, the value of the angle is $1.49252518$ .

$θ = 1.49252518$

Step 11

Substitute the values of $θ = 1.49252518$ and $| z | = \frac{\sqrt{2617}}{6}$ .

$\frac{\sqrt{2617}}{6 (\cos (1.49252518) + i \sin (1.49252518))}$