Trigonometry Examples

$(3 - 2 i) \cdot (7 + 6 i)$

Step 1

Expand $(3 - 2 i) (7 + 6 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$3 (7 + 6 i) - 2 i (7 + 6 i)$

Step 1.2

Apply the distributive property.

$3 \cdot 7 + 3 (6 i) - 2 i (7 + 6 i)$

Step 1.3

Apply the distributive property.

$3 \cdot 7 + 3 (6 i) - 2 i \cdot 7 - 2 i (6 i)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $3$ by $7$ .

$21 + 3 (6 i) - 2 i \cdot 7 - 2 i (6 i)$

Step 2.1.2

Multiply $6$ by $3$ .

$21 + 18 i - 2 i \cdot 7 - 2 i (6 i)$

Step 2.1.3

Multiply $7$ by $- 2$ .

$21 + 18 i - 14 i - 2 i (6 i)$

Step 2.1.4

Multiply $- 2 i (6 i)$ .

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Step 2.1.4.1

Multiply $6$ by $- 2$ .

$21 + 18 i - 14 i - 12 i i$

Step 2.1.4.2

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

$21 + 18 i - 14 i - 12 (i^{1} i)$

Step 2.1.4.3

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

$21 + 18 i - 14 i - 12 (i^{1} i^{1})$

Step 2.1.4.4

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

$21 + 18 i - 14 i - 12 i^{1 + 1}$

Step 2.1.4.5

Add $1$ and $1$ .

$21 + 18 i - 14 i - 12 i^{2}$

Step 2.1.5

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

$21 + 18 i - 14 i - 12 \cdot - 1$

Step 2.1.6

Multiply $- 12$ by $- 1$ .

$21 + 18 i - 14 i + 12$

Step 2.2

Add $21$ and $12$ .

$33 + 18 i - 14 i$

Step 2.3

Subtract $14 i$ from $18 i$ .

$33 + 4 i$

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 $a = 33$ and $b = 4$ .

$| z | = \sqrt{4^{2} + 33^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

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

$| z | = \sqrt{16 + 33^{2}}$

Step 6.2

Raise $33$ to the power of $2$ .

$| z | = \sqrt{16 + 1089}$

Step 6.3

Add $16$ and $1089$ .

$| z | = \sqrt{1105}$

Step 7

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

$θ = \arctan (\frac{4}{33})$

Step 8

Since inverse tangent of $\frac{4}{33}$ produces an angle in the first quadrant, the value of the angle is $0.12062366$ .

$θ = 0.12062366$

Step 9

Substitute the values of $θ = 0.12062366$ and $| z | = \sqrt{1105}$ .

$\sqrt{1105} (\cos (0.12062366) + i \sin (0.12062366))$