Trigonometry Examples

$32 + 32 i \sqrt{3}$ , $n = 3$

Step 1

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

$r = \sqrt{32^{2} + {(\sqrt{3} \cdot 32)}^{2}}$

Step 2

Simplify $\sqrt{32^{2} + {(\sqrt{3} \cdot 32)}^{2}}$ .

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Step 2.1

Simplify the expression.

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Step 2.1.1

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

$r = \sqrt{1024 + {(\sqrt{3} \cdot 32)}^{2}}$

Step 2.1.2

Move $32$ to the left of $\sqrt{3}$ .

$r = \sqrt{1024 + {(32 \cdot \sqrt{3})}^{2}}$

Step 2.1.3

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

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

Step 2.1.4

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

$r = \sqrt{1024 + 1024 {\sqrt{3}}^{2}}$

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

$r = \sqrt{1024 + 1024 {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$r = \sqrt{1024 + 1024 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$r = \sqrt{1024 + 1024 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$r = \sqrt{1024 + 1024 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$r = \sqrt{1024 + 1024 \cdot 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$r = \sqrt{1024 + 1024 \cdot 3}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $1024$ by $3$ .

$r = \sqrt{1024 + 3072}$

Step 2.3.2

Add $1024$ and $3072$ .

$r = \sqrt{4096}$

Step 2.3.3

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

$r = \sqrt{64^{2}}$

Step 2.3.4

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

$r = 64$

Step 3

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

$θ̂ = \arctan (| \frac{\sqrt{3} \cdot 32}{32} |)$

Step 4

Simplify $\arctan (| \frac{\sqrt{3} \cdot 32}{32} |)$ .

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Step 4.1

Cancel the common factor of $32$ .

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Step 4.1.1

Cancel the common factor.

$θ̂ = \arctan (| \frac{\sqrt{3} \cdot 32}{32} |)$

Step 4.1.2

Divide $\sqrt{3}$ by $1$ .

$θ̂ = \arctan (| \sqrt{3} |)$

Step 4.2

$\sqrt{3}$ is approximately $1.7320508$ which is positive so remove the absolute value

$θ̂ = \arctan (\sqrt{3})$

Step 4.3

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

$θ̂ = \frac{π}{3}$

Step 5

Find the quadrant.

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Step 5.1

Move $32$ to the left of $\sqrt{3}$ .

$(32, 32 \sqrt{3})$

Step 5.2

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. $θ = θ̂$

$θ = \frac{π}{3}$

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 ${(64)}^{\frac{1}{3}}$ and $\frac{(\frac{π}{3}) + 2 π k}{3}$ .

$c i s \frac{{(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k)}{3}$

Step 8.2

Combine $c$ and $\frac{{(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k)}{3}$ .

$i s \frac{c ({(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k))}{3}$

Step 8.3

Combine $i$ and $\frac{c ({(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k))}{3}$ .

$s \frac{i (c ({(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k)))}{3}$

Step 8.4

Combine $s$ and $\frac{i (c ({(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k)))}{3}$ .

$\frac{s (i (c ({(64)}^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k))))}{3}$

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

$\frac{s (i (c (64^{\frac{1}{3}} ((\frac{π}{3}) + 2 π k))))}{3}$

Step 8.5.2

Remove parentheses.

$\frac{s (i (c (64^{\frac{1}{3}} (\frac{π}{3} + 2 π k))))}{3}$

Step 8.5.3

Remove parentheses.

$\frac{s (i (c \cdot 64^{\frac{1}{3}} (\frac{π}{3} + 2 π k)))}{3}$

Step 8.5.4

Remove parentheses.

$\frac{s (i (c \cdot 64^{\frac{1}{3}}) (\frac{π}{3} + 2 π k))}{3}$

Step 8.5.5

Remove parentheses.

$\frac{s (i c \cdot 64^{\frac{1}{3}} (\frac{π}{3} + 2 π k))}{3}$

Step 8.5.6

Remove parentheses.

$\frac{s (i c \cdot 64^{\frac{1}{3}}) (\frac{π}{3} + 2 π k)}{3}$

Step 8.5.7

Remove parentheses.

$\frac{s (i c) \cdot 64^{\frac{1}{3}} (\frac{π}{3} + 2 π k)}{3}$

Step 8.5.8

Remove parentheses.

$\frac{s i c \cdot 64^{\frac{1}{3}} (\frac{π}{3} + 2 π k)}{3}$

Step 9

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

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Step 9.1

Rewrite $64$ as $4^{3}$ .

$k = 0 : {(4^{3})}^{\frac{1}{3}} c i s (\frac{(\frac{π}{3}) + 2 π (0)}{3})$

Step 9.2

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

$k = 0 : 4^{3 (\frac{1}{3})} c i s (\frac{(\frac{π}{3}) + 2 π (0)}{3})$

Step 9.3

Cancel the common factor of $3$ .

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Step 9.3.1

Cancel the common factor.

$k = 0 : 4^{3 (\frac{1}{3})} c i s (\frac{(\frac{π}{3}) + 2 π (0)}{3})$

Step 9.3.2

Rewrite the expression.

$k = 0 : 4 c i s (\frac{(\frac{π}{3}) + 2 π (0)}{3})$

Step 9.4

Evaluate the exponent.

$k = 0 : 4 c i s (\frac{(\frac{π}{3}) + 2 π (0)}{3})$

Step 9.5

Multiply $2 π (0)$ .

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Step 9.5.1

Multiply $0$ by $2$ .

$k = 0 : 4 c i s (\frac{\frac{π}{3} + 0 π}{3})$

Step 9.5.2

Multiply $0$ by $π$ .

$k = 0 : 4 c i s (\frac{\frac{π}{3} + 0}{3})$

Step 9.6

Add $\frac{π}{3}$ and $0$ .

$k = 0 : 4 c i s (\frac{\frac{π}{3}}{3})$

Step 9.7

Multiply the numerator by the reciprocal of the denominator.

$k = 0 : 4 c i s (\frac{π}{3} \cdot \frac{1}{3})$

Step 9.8

Multiply $\frac{π}{3} \cdot \frac{1}{3}$ .

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Step 9.8.1

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

$k = 0 : 4 c i s (\frac{π}{3 \cdot 3})$

Step 9.8.2

Multiply $3$ by $3$ .

$k = 0 : 4 c i s (\frac{π}{9})$

Step 10

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

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Step 10.1

Rewrite $64$ as $4^{3}$ .

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

Step 10.2

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

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

Step 10.3

Cancel the common factor of $3$ .

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Step 10.3.1

Cancel the common factor.

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

Step 10.3.2

Rewrite the expression.

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

Step 10.4

Evaluate the exponent.

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

Step 10.5

Multiply $2$ by $1$ .

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

Step 10.6

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.7

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 10.8

Combine the numerators over the common denominator.

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

Step 10.9

Simplify the numerator.

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Step 10.9.1

Multiply $3$ by $2$ .

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

Step 10.9.2

Add $π$ and $6 π$ .

$k = 1 : 4 c i s (\frac{\frac{7 π}{3}}{3})$

Step 10.10

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 4 c i s (\frac{7 π}{3} \cdot \frac{1}{3})$

Step 10.11

Multiply $\frac{7 π}{3} \cdot \frac{1}{3}$ .

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Step 10.11.1

Multiply $\frac{7 π}{3}$ by $\frac{1}{3}$ .

$k = 1 : 4 c i s (\frac{7 π}{3 \cdot 3})$

Step 10.11.2

Multiply $3$ by $3$ .

$k = 1 : 4 c i s (\frac{7 π}{9})$

Step 11

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

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Step 11.1

Rewrite $64$ as $4^{3}$ .

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

Step 11.2

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

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

Step 11.3

Cancel the common factor of $3$ .

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Step 11.3.1

Cancel the common factor.

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

Step 11.3.2

Rewrite the expression.

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

Step 11.4

Evaluate the exponent.

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

Step 11.5

Multiply $2$ by $2$ .

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

Step 11.6

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$k = 2 : 4 c i s (\frac{\frac{π}{3} + 4 π \cdot \frac{3}{3}}{3})$

Step 11.7

Combine $4 π$ and $\frac{3}{3}$ .

$k = 2 : 4 c i s (\frac{\frac{π}{3} + \frac{4 π \cdot 3}{3}}{3})$

Step 11.8

Combine the numerators over the common denominator.

$k = 2 : 4 c i s (\frac{\frac{π + 4 π \cdot 3}{3}}{3})$

Step 11.9

Simplify the numerator.

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Step 11.9.1

Multiply $3$ by $4$ .

$k = 2 : 4 c i s (\frac{\frac{π + 12 π}{3}}{3})$

Step 11.9.2

Add $π$ and $12 π$ .

$k = 2 : 4 c i s (\frac{\frac{13 π}{3}}{3})$

Step 11.10

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : 4 c i s (\frac{13 π}{3} \cdot \frac{1}{3})$

Step 11.11

Multiply $\frac{13 π}{3} \cdot \frac{1}{3}$ .

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Step 11.11.1

Multiply $\frac{13 π}{3}$ by $\frac{1}{3}$ .

$k = 2 : 4 c i s (\frac{13 π}{3 \cdot 3})$

Step 11.11.2

Multiply $3$ by $3$ .

$k = 2 : 4 c i s (\frac{13 π}{9})$

Step 12

List the solutions.

$k = 0 : 4 c i s (\frac{π}{9})$

$k = 1 : 4 c i s (\frac{7 π}{9})$

$k = 2 : 4 c i s (\frac{13 π}{9})$

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