Trigonometry Examples

- 4 - 4 i \sqrt{3} + 4 i - 4 \sqrt{3}

$- 4 - 4 i \sqrt{3} + 4 i - 4 \sqrt{3}$

Step 1

Reorder

- 4 - 4 i \sqrt{3} + 4 i

$- 4 - 4 i \sqrt{3} + 4 i$ and

- 4 \sqrt{3}

$- 4 \sqrt{3}$ .

- 4 \sqrt{3} - 4 - 4 i \sqrt{3} + 4 i

$- 4 \sqrt{3} - 4 - 4 i \sqrt{3} + 4 i$

Step 2

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

z = a + b i = | z | (cos (θ) + i sin (θ))

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 3

The modulus of a complex number is the distance from the origin on the complex plane.

| z | = \sqrt{a^{2} + b^{2}}

$| z | = \sqrt{a^{2} + b^{2}}$ where

z = a + b i

$z = a + b i$

Step 4

Substitute the actual values of

a = - 4 \sqrt{3}

$a = - 4 \sqrt{3}$ and

b = - 4

$b = - 4$ .

| z | = \sqrt{{(- 4)}^{2} + {(- 4 \sqrt{3})}^{2}}

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

Step 5

Find

| z |

$| z |$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Raise

- 4

$- 4$ to the power of

2

$2$ .

| z | = \sqrt{16 + {(- 4 \sqrt{3})}^{2}}

$| z | = \sqrt{16 + {(- 4 \sqrt{3})}^{2}}$

Step 5.1.2

Apply the product rule to

- 4 \sqrt{3}

$- 4 \sqrt{3}$ .

| z | = \sqrt{16 + {(- 4)}^{2} {\sqrt{3}}^{2}}

$| z | = \sqrt{16 + {(- 4)}^{2} {\sqrt{3}}^{2}}$

Step 5.1.3

Raise

- 4

$- 4$ to the power of

2

$2$ .

| z | = \sqrt{16 + 16 {\sqrt{3}}^{2}}

$| z | = \sqrt{16 + 16 {\sqrt{3}}^{2}}$

| z | = \sqrt{16 + 16 {\sqrt{3}}^{2}}

$| z | = \sqrt{16 + 16 {\sqrt{3}}^{2}}$

Step 5.2

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 5.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

| z | = \sqrt{16 + 16 {(3^{\frac{1}{2}})}^{2}}

$| z | = \sqrt{16 + 16 {(3^{\frac{1}{2}})}^{2}}$

Step 5.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

| z | = \sqrt{16 + 16 \cdot 3^{\frac{1}{2} \cdot 2}}

$| z | = \sqrt{16 + 16 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

| z | = \sqrt{16 + 16 \cdot 3^{\frac{2}{2}}}

$| z | = \sqrt{16 + 16 \cdot 3^{\frac{2}{2}}}$

Step 5.2.4

Cancel the common factor of

2

$2$ .

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Step 5.2.4.1

Cancel the common factor.

$| z | = \sqrt{16 + 16 \cdot 3^{\frac{2}{2}}}$

Step 5.2.4.2

Rewrite the expression.

$| z | = \sqrt{16 + 16 \cdot 3}$

Step 5.2.5

Evaluate the exponent.

$| z | = \sqrt{16 + 16 \cdot 3}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply

$16$ by

$3$ .

$| z | = \sqrt{16 + 48}$

Step 5.3.2

Add

$16$ and

$48$ .

$| z | = \sqrt{64}$

Step 5.3.3

Rewrite

$64$ as

$8^{2}$ .

$| z | = \sqrt{8^{2}}$

Step 5.3.4

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

$| z | = 8$

Step 6

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

$θ = \arctan (\frac{- 4}{- 4 \sqrt{3}})$

Step 7

Since inverse tangent of

$\frac{- 4}{- 4 \sqrt{3}}$ produces an angle in the third quadrant, the value of the angle is

$\frac{7 π}{6}$ .

$θ = \frac{7 π}{6}$

Step 8

Substitute the values of

$θ = \frac{7 π}{6}$ and

$| z | = 8$ .

$8 (\cos (\frac{7 π}{6}) + i \sin (\frac{7 π}{6}))$