Trigonometry Examples

$(4 \sqrt{3} - 4 i) \cdot (8 i)$

Step 1

Apply the distributive property.

$4 \sqrt{3} (8 i) - 4 i (8 i)$

Step 2

Multiply $8$ by $4$ .

$32 \sqrt{3} i - 4 i (8 i)$

Step 3

Multiply $- 4 i (8 i)$ .

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Step 3.1

Multiply $8$ by $- 4$ .

$32 \sqrt{3} i - 32 i i$

Step 3.2

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

$32 \sqrt{3} i - 32 (i^{1} i)$

Step 3.3

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

$32 \sqrt{3} i - 32 (i^{1} i^{1})$

Step 3.4

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

$32 \sqrt{3} i - 32 i^{1 + 1}$

Step 3.5

Add $1$ and $1$ .

$32 \sqrt{3} i - 32 i^{2}$

Step 4

Simplify each term.

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Step 4.1

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

$32 \sqrt{3} i - 32 \cdot - 1$

Step 4.2

Multiply $- 32$ by $- 1$ .

$32 \sqrt{3} i + 32$

Step 5

Reorder $32 \sqrt{3} i$ and $32$ .

$32 + 32 \sqrt{3} i$

Step 6

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 7

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 8

Substitute the actual values of $a = 32$ and $b = 32 \sqrt{3}$ .

$| z | = \sqrt{{(32 \sqrt{3})}^{2} + 32^{2}}$

Step 9

Find $| z |$ .

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Step 9.1

Simplify the expression.

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Step 9.1.1

Apply the product rule to $32 \sqrt{3}$ .

$| z | = \sqrt{32^{2} {\sqrt{3}}^{2} + 32^{2}}$

Step 9.1.2

Raise $32$ to the power of $2$ .

$| z | = \sqrt{1024 {\sqrt{3}}^{2} + 32^{2}}$

Step 9.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \sqrt{1024 {(3^{\frac{1}{2}})}^{2} + 32^{2}}$

Step 9.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{1024 \cdot 3^{\frac{1}{2} \cdot 2} + 32^{2}}$

Step 9.2.3

Combine $\frac{1}{2}$ and $2$ .

$| z | = \sqrt{1024 \cdot 3^{\frac{2}{2}} + 32^{2}}$

Step 9.2.4

Cancel the common factor of $2$ .

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Step 9.2.4.1

Cancel the common factor.

$| z | = \sqrt{1024 \cdot 3^{\frac{2}{2}} + 32^{2}}$

Step 9.2.4.2

Rewrite the expression.

$| z | = \sqrt{1024 \cdot 3 + 32^{2}}$

Step 9.2.5

Evaluate the exponent.

$| z | = \sqrt{1024 \cdot 3 + 32^{2}}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Multiply $1024$ by $3$ .

$| z | = \sqrt{3072 + 32^{2}}$

Step 9.3.2

Raise $32$ to the power of $2$ .

$| z | = \sqrt{3072 + 1024}$

Step 9.3.3

Add $3072$ and $1024$ .

$| z | = \sqrt{4096}$

Step 9.3.4

Rewrite $4096$ as $64^{2}$ .

$| z | = \sqrt{64^{2}}$

Step 9.3.5

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

$| z | = 64$

Step 10

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

$θ = \arctan (\frac{32 \sqrt{3}}{32})$

Step 11

Since inverse tangent of $\frac{32 \sqrt{3}}{32}$ produces an angle in the first quadrant, the value of the angle is $\frac{π}{3}$ .

$θ = \frac{π}{3}$

Step 12

Substitute the values of $θ = \frac{π}{3}$ and $| z | = 64$ .

$64 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))$