Precalculus Examples

$(5 + 5 i) (8 (\cos (\frac{3 π}{7}) + i \sin (\frac{3 π}{7})))$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Evaluate $\cos (\frac{3 π}{7})$ .

$(5 + 5 i) (8 (0.22252093 + i \sin (\frac{3 π}{7})))$

Step 1.1.2

Evaluate $\sin (\frac{3 π}{7})$ .

$(5 + 5 i) (8 (0.22252093 + i \cdot 0.97492791))$

Step 1.1.3

Move $0.97492791$ to the left of $i$ .

$(5 + 5 i) (8 (0.22252093 + 0.97492791 i))$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$(5 + 5 i) (8 \cdot 0.22252093 + 8 (0.97492791 i))$

Step 1.2.2

Multiply.

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Step 1.2.2.1

Multiply $8$ by $0.22252093$ .

$(5 + 5 i) (1.78016747 + 8 (0.97492791 i))$

Step 1.2.2.2

Multiply $0.97492791$ by $8$ .

$(5 + 5 i) (1.78016747 + 7.79942329 i)$

Step 2

Expand $(5 + 5 i) (1.78016747 + 7.79942329 i)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$5 (1.78016747 + 7.79942329 i) + 5 i (1.78016747 + 7.79942329 i)$

Step 2.2

Apply the distributive property.

$5 \cdot 1.78016747 + 5 (7.79942329 i) + 5 i (1.78016747 + 7.79942329 i)$

Step 2.3

Apply the distributive property.

$5 \cdot 1.78016747 + 5 (7.79942329 i) + 5 i \cdot 1.78016747 + 5 i (7.79942329 i)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $5$ by $1.78016747$ .

$8.90083735 + 5 (7.79942329 i) + 5 i \cdot 1.78016747 + 5 i (7.79942329 i)$

Step 3.1.2

Multiply $7.79942329$ by $5$ .

$8.90083735 + 38.99711648 i + 5 i \cdot 1.78016747 + 5 i (7.79942329 i)$

Step 3.1.3

Multiply $1.78016747$ by $5$ .

$8.90083735 + 38.99711648 i + 8.90083735 i + 5 i (7.79942329 i)$

Step 3.1.4

Multiply $5 i (7.79942329 i)$ .

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Step 3.1.4.1

Multiply $7.79942329$ by $5$ .

$8.90083735 + 38.99711648 i + 8.90083735 i + 38.99711648 i i$

Step 3.1.4.2

Raise $i$ to the power of $1$ .

$8.90083735 + 38.99711648 i + 8.90083735 i + 38.99711648 (i^{1} i)$

Step 3.1.4.3

Raise $i$ to the power of $1$ .

$8.90083735 + 38.99711648 i + 8.90083735 i + 38.99711648 (i^{1} i^{1})$

Step 3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8.90083735 + 38.99711648 i + 8.90083735 i + 38.99711648 i^{1 + 1}$

Step 3.1.4.5

Add $1$ and $1$ .

$8.90083735 + 38.99711648 i + 8.90083735 i + 38.99711648 i^{2}$

Step 3.1.5

Rewrite $i^{2}$ as $- 1$ .

$8.90083735 + 38.99711648 i + 8.90083735 i + 38.99711648 \cdot - 1$

Step 3.1.6

Multiply $38.99711648$ by $- 1$ .

$8.90083735 + 38.99711648 i + 8.90083735 i - 38.99711648$

Step 3.2

Subtract $38.99711648$ from $8.90083735$ .

$- 30.09627912 + 38.99711648 i + 8.90083735 i$

Step 3.3

Add $38.99711648 i$ and $8.90083735 i$ .

$- 30.09627912 + 47.89795384 i$

Step 4

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 5

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 6

Substitute the actual values of $a = - 30.09627912$ and $b = 47.89795384$ .

$| z | = \sqrt{{47.89795384}^{2} + {(- 30.09627912)}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Raise $47.89795384$ to the power of $2$ .

$| z | = \sqrt{2294.21398258 + {(- 30.09627912)}^{2}}$

Step 7.2

Raise $- 30.09627912$ to the power of $2$ .

$| z | = \sqrt{2294.21398258 + 905.78601741}$

Step 7.3

Add $2294.21398258$ and $905.78601741$ .

$| z | = \sqrt{3199.\overline{9}}$

Step 8

Evaluate the root.

$| z | = 56.56854249$

Step 9

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

$θ = \arctan (\frac{47.89795384}{- 30.09627912})$

Step 10

Since inverse tangent of $\frac{47.89795384}{- 30.09627912}$ produces an angle in the second quadrant, the value of the angle is $2.13179501$ .

$θ = 2.13179501$

Step 11

Substitute the values of $θ = 2.13179501$ and $| z | = 56.56854249$ .

$56.56854249 (\cos (2.13179501) + i \sin (2.13179501))$