Trigonometry Examples

$16 \sqrt{3} + 16 i$ , $n = 4$

Step 1

Calculate the distance from $(a, b)$ to the origin using the formula $r = \sqrt{a^{2} + b^{2}}$ .

$r = \sqrt{{(16 \sqrt{3})}^{2} + 16^{2}}$

Step 2

Simplify $\sqrt{{(16 \sqrt{3})}^{2} + 16^{2}}$ .

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the product rule to $16 \sqrt{3}$ .

$r = \sqrt{16^{2} {\sqrt{3}}^{2} + 16^{2}}$

Step 2.1.2

Raise $16$ to the power of $2$ .

$r = \sqrt{256 {\sqrt{3}}^{2} + 16^{2}}$

Step 2.2

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

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Step 2.2.1

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

$r = \sqrt{256 {(3^{\frac{1}{2}})}^{2} + 16^{2}}$

Step 2.2.2

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

$r = \sqrt{256 \cdot 3^{\frac{1}{2} \cdot 2} + 16^{2}}$

Step 2.2.3

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

$r = \sqrt{256 \cdot 3^{\frac{2}{2}} + 16^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$r = \sqrt{256 \cdot 3^{\frac{2}{2}} + 16^{2}}$

Step 2.2.4.2

Rewrite the expression.

$r = \sqrt{256 \cdot 3^{1} + 16^{2}}$

Step 2.2.5

Evaluate the exponent.

$r = \sqrt{256 \cdot 3 + 16^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $256$ by $3$ .

$r = \sqrt{768 + 16^{2}}$

Step 2.3.2

Raise $16$ to the power of $2$ .

$r = \sqrt{768 + 256}$

Step 2.3.3

Add $768$ and $256$ .

$r = \sqrt{1024}$

Step 2.3.4

Rewrite $1024$ as $32^{2}$ .

$r = \sqrt{32^{2}}$

Step 2.3.5

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

$r = 32$

Step 3

Calculate reference angle $θ̂ = \arctan (| \frac{b}{a} |)$ .

$θ̂ = \arctan (| \frac{16}{16 \sqrt{3}} |)$

Step 4

Simplify $\arctan (| \frac{16}{16 \sqrt{3}} |)$ .

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Step 4.1

Cancel the common factor of $16$ .

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Step 4.1.1

Cancel the common factor.

$θ̂ = \arctan (| \frac{16}{16 \sqrt{3}} |)$

Step 4.1.2

Rewrite the expression.

$θ̂ = \arctan (| \frac{1}{\sqrt{3}} |)$

Step 4.2

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$θ̂ = \arctan (| \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} |)$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$θ̂ = \arctan (| \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} |)$

Step 4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$θ̂ = \arctan (| \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} |)$

Step 4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$θ̂ = \arctan (| \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} |)$

Step 4.3.4

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

$θ̂ = \arctan (| \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} |)$

Step 4.3.5

Add $1$ and $1$ .

$θ̂ = \arctan (| \frac{\sqrt{3}}{{\sqrt{3}}^{2}} |)$

Step 4.3.6

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

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Step 4.3.6.1

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

$θ̂ = \arctan (| \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} |)$

Step 4.3.6.2

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

$θ̂ = \arctan (| \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} |)$

Step 4.3.6.3

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

$θ̂ = \arctan (| \frac{\sqrt{3}}{3^{\frac{2}{2}}} |)$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$θ̂ = \arctan (| \frac{\sqrt{3}}{3^{\frac{2}{2}}} |)$

Step 4.3.6.4.2

Rewrite the expression.

$θ̂ = \arctan (| \frac{\sqrt{3}}{3^{1}} |)$

Step 4.3.6.5

Evaluate the exponent.

$θ̂ = \arctan (| \frac{\sqrt{3}}{3} |)$

Step 4.4

$\frac{\sqrt{3}}{3}$ is approximately $0.57735026$ which is positive so remove the absolute value

$θ̂ = \arctan (\frac{\sqrt{3}}{3})$

Step 4.5

The exact value of $\arctan (\frac{\sqrt{3}}{3})$ is $\frac{π}{6}$ .

$θ̂ = \frac{π}{6}$

Step 5

The point is located in the first quadrant because $x$ and $y$ are both positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Quadrant $1$

Step 6

$(a, b)$ is in the first quadrant. $θ = θ̂$

$θ = \frac{π}{6}$

Step 7

Use the formula to find the roots of the complex number.

${(a + b i)}^{\frac{1}{n}} = r^{\frac{1}{n}} c i s (\frac{θ + 2 π k}{n})$ , $k = 0, 1, \dots, n - 1$

Step 8

Substitute $r$ , $n$ , and $θ$ into the formula.

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Step 8.1

Combine ${(32)}^{\frac{1}{4}}$ and $\frac{(\frac{π}{6}) + 2 π k}{4}$ .

$c i s \frac{{(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k)}{4}$

Step 8.2

Combine $c$ and $\frac{{(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k)}{4}$ .

$i s \frac{c ({(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k))}{4}$

Step 8.3

Combine $i$ and $\frac{c ({(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k))}{4}$ .

$s \frac{i (c ({(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k)))}{4}$

Step 8.4

Combine $s$ and $\frac{i (c ({(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k)))}{4}$ .

$\frac{s (i (c ({(32)}^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k))))}{4}$

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

$\frac{s (i (c (32^{\frac{1}{4}} ((\frac{π}{6}) + 2 π k))))}{4}$

Step 8.5.2

Remove parentheses.

$\frac{s (i (c (32^{\frac{1}{4}} (\frac{π}{6} + 2 π k))))}{4}$

Step 8.5.3

Remove parentheses.

$\frac{s (i (c \cdot 32^{\frac{1}{4}} (\frac{π}{6} + 2 π k)))}{4}$

Step 8.5.4

Remove parentheses.

$\frac{s (i (c \cdot 32^{\frac{1}{4}}) (\frac{π}{6} + 2 π k))}{4}$

Step 8.5.5

Remove parentheses.

$\frac{s (i c \cdot 32^{\frac{1}{4}} (\frac{π}{6} + 2 π k))}{4}$

Step 8.5.6

Remove parentheses.

$\frac{s (i c \cdot 32^{\frac{1}{4}}) (\frac{π}{6} + 2 π k)}{4}$

Step 8.5.7

Remove parentheses.

$\frac{s (i c) \cdot 32^{\frac{1}{4}} (\frac{π}{6} + 2 π k)}{4}$

Step 8.5.8

Remove parentheses.

$\frac{s i c \cdot 32^{\frac{1}{4}} (\frac{π}{6} + 2 π k)}{4}$

Step 9

Substitute $k = 0$ into the formula and simplify.

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Step 9.1

Remove parentheses.

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{(\frac{π}{6}) + 2 π (0)}{4})$

Step 9.2

Multiply $2 π (0)$ .

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Step 9.2.1

Multiply $0$ by $2$ .

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 0 π}{4})$

Step 9.2.2

Multiply $0$ by $π$ .

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 0}{4})$

Step 9.3

Add $\frac{π}{6}$ and $0$ .

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6}}{4})$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{π}{6} \cdot \frac{1}{4})$

Step 9.5

Multiply $\frac{π}{6} \cdot \frac{1}{4}$ .

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Step 9.5.1

Multiply $\frac{π}{6}$ by $\frac{1}{4}$ .

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{π}{6 \cdot 4})$

Step 9.5.2

Multiply $6$ by $4$ .

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{π}{24})$

Step 10

Substitute $k = 1$ into the formula and simplify.

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Step 10.1

Remove parentheses.

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{(\frac{π}{6}) + 2 π (1)}{4})$

Step 10.2

Multiply $2$ by $1$ .

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 2 π}{4})$

Step 10.3

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

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 2 π \cdot \frac{6}{6}}{4})$

Step 10.4

Combine $2 π$ and $\frac{6}{6}$ .

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + \frac{2 π \cdot 6}{6}}{4})$

Step 10.5

Combine the numerators over the common denominator.

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{\frac{π + 2 π \cdot 6}{6}}{4})$

Step 10.6

Simplify the numerator.

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Step 10.6.1

Multiply $6$ by $2$ .

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{\frac{π + 12 π}{6}}{4})$

Step 10.6.2

Add $π$ and $12 π$ .

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{\frac{13 π}{6}}{4})$

Step 10.7

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{13 π}{6} \cdot \frac{1}{4})$

Step 10.8

Multiply $\frac{13 π}{6} \cdot \frac{1}{4}$ .

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Step 10.8.1

Multiply $\frac{13 π}{6}$ by $\frac{1}{4}$ .

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{13 π}{6 \cdot 4})$

Step 10.8.2

Multiply $6$ by $4$ .

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{13 π}{24})$

Step 11

Substitute $k = 2$ into the formula and simplify.

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Step 11.1

Remove parentheses.

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{(\frac{π}{6}) + 2 π (2)}{4})$

Step 11.2

Multiply $2$ by $2$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 4 π}{4})$

Step 11.3

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 4 π \cdot \frac{6}{6}}{4})$

Step 11.4

Combine $4 π$ and $\frac{6}{6}$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + \frac{4 π \cdot 6}{6}}{4})$

Step 11.5

Combine the numerators over the common denominator.

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{\frac{π + 4 π \cdot 6}{6}}{4})$

Step 11.6

Simplify the numerator.

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Step 11.6.1

Multiply $6$ by $4$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{\frac{π + 24 π}{6}}{4})$

Step 11.6.2

Add $π$ and $24 π$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{\frac{25 π}{6}}{4})$

Step 11.7

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{25 π}{6} \cdot \frac{1}{4})$

Step 11.8

Multiply $\frac{25 π}{6} \cdot \frac{1}{4}$ .

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Step 11.8.1

Multiply $\frac{25 π}{6}$ by $\frac{1}{4}$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{25 π}{6 \cdot 4})$

Step 11.8.2

Multiply $6$ by $4$ .

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{25 π}{24})$

Step 12

Substitute $k = 3$ into the formula and simplify.

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Step 12.1

Remove parentheses.

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{(\frac{π}{6}) + 2 π (3)}{4})$

Step 12.2

Multiply $3$ by $2$ .

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 6 π}{4})$

Step 12.3

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

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + 6 π \cdot \frac{6}{6}}{4})$

Step 12.4

Combine $6 π$ and $\frac{6}{6}$ .

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{\frac{π}{6} + \frac{6 π \cdot 6}{6}}{4})$

Step 12.5

Combine the numerators over the common denominator.

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{\frac{π + 6 π \cdot 6}{6}}{4})$

Step 12.6

Simplify the numerator.

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Step 12.6.1

Multiply $6$ by $6$ .

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{\frac{π + 36 π}{6}}{4})$

Step 12.6.2

Add $π$ and $36 π$ .

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{\frac{37 π}{6}}{4})$

Step 12.7

Multiply the numerator by the reciprocal of the denominator.

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{37 π}{6} \cdot \frac{1}{4})$

Step 12.8

Multiply $\frac{37 π}{6} \cdot \frac{1}{4}$ .

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Step 12.8.1

Multiply $\frac{37 π}{6}$ by $\frac{1}{4}$ .

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{37 π}{6 \cdot 4})$

Step 12.8.2

Multiply $6$ by $4$ .

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{37 π}{24})$

Step 13

List the solutions.

$k = 0 : 32^{\frac{1}{4}} c i s (\frac{π}{24})$

$k = 1 : 32^{\frac{1}{4}} c i s (\frac{13 π}{24})$

$k = 2 : 32^{\frac{1}{4}} c i s (\frac{25 π}{24})$

$k = 3 : 32^{\frac{1}{4}} c i s (\frac{37 π}{24})$

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