Trigonometry Examples

- \frac{27 \sqrt{2}}{2} + \frac{27 \sqrt{2}}{2} i

$- \frac{27 \sqrt{2}}{2} + \frac{27 \sqrt{2}}{2} i$ ,

n = 3

$n = 3$

Step 1

Calculate the distance from

(a, b)

$(a, b)$ to the origin using the formula

r = \sqrt{a^{2} + b^{2}}

$r = \sqrt{a^{2} + b^{2}}$ .

r = \sqrt{{(- \frac{27 \sqrt{2}}{2})}^{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{{(- \frac{27 \sqrt{2}}{2})}^{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2

Simplify

\sqrt{{(- \frac{27 \sqrt{2}}{2})}^{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$\sqrt{{(- \frac{27 \sqrt{2}}{2})}^{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}$ .

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Step 2.1

Use the power rule

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

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to

- \frac{27 \sqrt{2}}{2}

$- \frac{27 \sqrt{2}}{2}$ .

r = \sqrt{{(- 1)}^{2} {(\frac{27 \sqrt{2}}{2})}^{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{{(- 1)}^{2} {(\frac{27 \sqrt{2}}{2})}^{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.1.2

Apply the product rule to

\frac{27 \sqrt{2}}{2}

$\frac{27 \sqrt{2}}{2}$ .

r = \sqrt{{(- 1)}^{2} \frac{{(27 \sqrt{2})}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{{(- 1)}^{2} \frac{{(27 \sqrt{2})}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.1.3

Apply the product rule to

27 \sqrt{2}

$27 \sqrt{2}$ .

r = \sqrt{{(- 1)}^{2} \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{{(- 1)}^{2} \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

r = \sqrt{{(- 1)}^{2} \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{{(- 1)}^{2} \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

r = \sqrt{1 \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{1 \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.2.2

Multiply

\frac{27^{2} {\sqrt{2}}^{2}}{2^{2}}

$\frac{27^{2} {\sqrt{2}}^{2}}{2^{2}}$ by

1

$1$ .

r = \sqrt{\frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{\frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

r = \sqrt{\frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{\frac{27^{2} {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Raise

27

$27$ to the power of

2

$2$ .

r = \sqrt{\frac{729 {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{\frac{729 {\sqrt{2}}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3.2

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 2.3.2.1

Use

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

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

\sqrt{2}

$\sqrt{2}$ as

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

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

r = \sqrt{\frac{729 {(2^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{\frac{729 {(2^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3.2.2

Apply the power rule and multiply exponents,

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

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

r = \sqrt{\frac{729 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}

$r = \sqrt{\frac{729 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$r = \sqrt{\frac{729 \cdot 2^{\frac{2}{2}}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3.2.4

Cancel the common factor of

$2$ .

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Step 2.3.2.4.1

Cancel the common factor.

$r = \sqrt{\frac{729 \cdot 2^{\frac{2}{2}}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3.2.4.2

Rewrite the expression.

$r = \sqrt{\frac{729 \cdot 2^{1}}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.3.2.5

Evaluate the exponent.

$r = \sqrt{\frac{729 \cdot 2}{2^{2}} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.4

Reduce the expression by cancelling the common factors.

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Step 2.4.1

Raise

$2$ to the power of

$2$ .

$r = \sqrt{\frac{729 \cdot 2}{4} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.4.2

Multiply

$729$ by

$2$ .

$r = \sqrt{\frac{1458}{4} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.4.3

Cancel the common factor of

$1458$ and

$4$ .

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Step 2.4.3.1

Factor

$2$ out of

$1458$ .

$r = \sqrt{\frac{2 (729)}{4} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.4.3.2

Cancel the common factors.

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Step 2.4.3.2.1

Factor

$2$ out of

$4$ .

$r = \sqrt{\frac{2 \cdot 729}{2 \cdot 2} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.4.3.2.2

Cancel the common factor.

$r = \sqrt{\frac{2 \cdot 729}{2 \cdot 2} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.4.3.2.3

Rewrite the expression.

$r = \sqrt{\frac{729}{2} + {(\frac{27 \sqrt{2}}{2})}^{2}}$

Step 2.5

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.5.1

Apply the product rule to

$\frac{27 \sqrt{2}}{2}$ .

$r = \sqrt{\frac{729}{2} + \frac{{(27 \sqrt{2})}^{2}}{2^{2}}}$

Step 2.5.2

Apply the product rule to

$27 \sqrt{2}$ .

$r = \sqrt{\frac{729}{2} + \frac{27^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Raise

$27$ to the power of

$2$ .

$r = \sqrt{\frac{729}{2} + \frac{729 {\sqrt{2}}^{2}}{2^{2}}}$

Step 2.6.2

Rewrite

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

$2$ .

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Step 2.6.2.1

Use

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

$\sqrt{2}$ as

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

$r = \sqrt{\frac{729}{2} + \frac{729 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 2.6.2.2

Apply the power rule and multiply exponents,

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

$r = \sqrt{\frac{729}{2} + \frac{729 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.6.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$r = \sqrt{\frac{729}{2} + \frac{729 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.2.4

Cancel the common factor of

$2$ .

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Step 2.6.2.4.1

Cancel the common factor.

$r = \sqrt{\frac{729}{2} + \frac{729 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.2.4.2

Rewrite the expression.

$r = \sqrt{\frac{729}{2} + \frac{729 \cdot 2^{1}}{2^{2}}}$

Step 2.6.2.5

Evaluate the exponent.

$r = \sqrt{\frac{729}{2} + \frac{729 \cdot 2}{2^{2}}}$

Step 2.7

Reduce the expression by cancelling the common factors.

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Step 2.7.1

Raise

$2$ to the power of

$2$ .

$r = \sqrt{\frac{729}{2} + \frac{729 \cdot 2}{4}}$

Step 2.7.2

Multiply

$729$ by

$2$ .

$r = \sqrt{\frac{729}{2} + \frac{1458}{4}}$

Step 2.7.3

Cancel the common factor of

$1458$ and

$4$ .

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Step 2.7.3.1

Factor

$2$ out of

$1458$ .

$r = \sqrt{\frac{729}{2} + \frac{2 (729)}{4}}$

Step 2.7.3.2

Cancel the common factors.

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Step 2.7.3.2.1

Factor

$2$ out of

$4$ .

$r = \sqrt{\frac{729}{2} + \frac{2 \cdot 729}{2 \cdot 2}}$

Step 2.7.3.2.2

Cancel the common factor.

$r = \sqrt{\frac{729}{2} + \frac{2 \cdot 729}{2 \cdot 2}}$

Step 2.7.3.2.3

Rewrite the expression.

$r = \sqrt{\frac{729}{2} + \frac{729}{2}}$

Step 2.7.4

Simplify the expression.

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Step 2.7.4.1

Combine the numerators over the common denominator.

$r = \sqrt{\frac{729 + 729}{2}}$

Step 2.7.4.2

Add

$729$ and

$729$ .

$r = \sqrt{\frac{1458}{2}}$

Step 2.7.4.3

Divide

$1458$ by

$2$ .

$r = \sqrt{729}$

Step 2.7.4.4

Rewrite

$729$ as

$27^{2}$ .

$r = \sqrt{27^{2}}$

Step 2.7.4.5

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

$r = 27$

Step 3

Calculate reference angle

$θ̂ = \arctan (| \frac{b}{a} |)$ .

$θ̂ = \arctan (| \frac{\frac{27 \sqrt{2}}{2}}{- \frac{27 \sqrt{2}}{2}} |)$

Step 4

Simplify

$\arctan (| \frac{\frac{27 \sqrt{2}}{2}}{- \frac{27 \sqrt{2}}{2}} |)$ .

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Step 4.1

Cancel the common factor of

$\frac{27 \sqrt{2}}{2}$ .

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Step 4.1.1

Cancel the common factor.

$θ̂ = \arctan (| \frac{\frac{27 \sqrt{2}}{2}}{- \frac{27 \sqrt{2}}{2}} |)$

Step 4.1.2

Rewrite the expression.

$θ̂ = \arctan (| \frac{1}{- 1} |)$

Step 4.1.3

Move the negative one from the denominator of

$\frac{1}{- 1}$ .

$θ̂ = \arctan (| - 1 \cdot 1 |)$

Step 4.2

Multiply

$- 1$ by

$1$ .

$θ̂ = \arctan (| - 1 |)$

Step 4.3

The absolute value is the distance between a number and zero. The distance between

$- 1$ and

$0$ is

$1$ .

$θ̂ = \arctan (1)$

Step 4.4

The exact value of

$\arctan (1)$ is

$\frac{π}{4}$ .

$θ̂ = \frac{π}{4}$

Step 5

The point is located in the second quadrant because

$x$ is negative and

$y$ is positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Quadrant

$2$

Step 6

$(a, b)$ is in the second quadrant.

$θ = π - θ̂$

$θ = π - \frac{π}{4}$

Step 7

Simplify

$θ$ .

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Step 7.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine

$π$ and

$\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 7.2.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Move

$4$ to the left of

$π$ .

$\frac{4 \cdot π - π}{4}$

Step 7.3.2

Subtract

$π$ from

$4 π$ .

$\frac{3 π}{4}$

Step 8

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 9

Substitute

$r$ ,

$n$ , and

$θ$ into the formula.

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Step 9.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

${(27)}^{\frac{1}{3}} c i s \frac{π \cdot \frac{4}{4} - \frac{π}{4} + 2 π k}{3}$

Step 9.2

Combine

$π$ and

$\frac{4}{4}$ .

${(27)}^{\frac{1}{3}} c i s \frac{\frac{π \cdot 4}{4} - \frac{π}{4} + 2 π k}{3}$

Step 9.3

Combine the numerators over the common denominator.

${(27)}^{\frac{1}{3}} c i s \frac{\frac{π \cdot 4 - π}{4} + 2 π k}{3}$

Step 9.4

Subtract

$π$ from

$π \cdot 4$ .

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Step 9.4.1

Reorder

$π$ and

$4$ .

${(27)}^{\frac{1}{3}} c i s \frac{\frac{4 \cdot π - π}{4} + 2 π k}{3}$

Step 9.4.2

Subtract

$π$ from

$4 \cdot π$ .

${(27)}^{\frac{1}{3}} c i s \frac{\frac{3 \cdot π}{4} + 2 π k}{3}$

Step 9.5

Combine

${(27)}^{\frac{1}{3}}$ and

$\frac{\frac{3 \cdot π}{4} + 2 π k}{3}$ .

$c i s \frac{{(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)}{3}$

Step 9.6

Combine

$c$ and

$\frac{{(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)}{3}$ .

$i s \frac{c ({(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k))}{3}$

Step 9.7

Combine

$i$ and

$\frac{c ({(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k))}{3}$ .

$s \frac{i (c ({(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)))}{3}$

Step 9.8

Combine

$s$ and

$\frac{i (c ({(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)))}{3}$ .

$\frac{s (i (c ({(27)}^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k))))}{3}$

Step 9.9

Remove parentheses.

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Step 9.9.1

Remove parentheses.

$\frac{s (i (c (27^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k))))}{3}$

Step 9.9.2

Remove parentheses.

$\frac{s (i (c \cdot 27^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)))}{3}$

Step 9.9.3

Remove parentheses.

$\frac{s (i (c \cdot 27^{\frac{1}{3}}) (\frac{3 \cdot π}{4} + 2 π k))}{3}$

Step 9.9.4

Remove parentheses.

$\frac{s (i c \cdot 27^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k))}{3}$

Step 9.9.5

Remove parentheses.

$\frac{s (i c \cdot 27^{\frac{1}{3}}) (\frac{3 \cdot π}{4} + 2 π k)}{3}$

Step 9.9.6

Remove parentheses.

$\frac{s (i c) \cdot 27^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)}{3}$

Step 9.9.7

Remove parentheses.

$\frac{s i c \cdot 27^{\frac{1}{3}} (\frac{3 \cdot π}{4} + 2 π k)}{3}$

Step 10

Substitute

$k = 0$ into the formula and simplify.

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Step 10.1

Rewrite

$27$ as

$3^{3}$ .

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

Step 10.2

Apply the power rule and multiply exponents,

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

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

Step 10.3

Cancel the common factor of

$3$ .

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Step 10.3.1

Cancel the common factor.

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

Step 10.3.2

Rewrite the expression.

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

Step 10.4

Evaluate the exponent.

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

Step 10.5

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 10.6

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 10.7

Combine the numerators over the common denominator.

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

Step 10.8

Simplify the numerator.

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Step 10.8.1

Move

$4$ to the left of

$π$ .

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

Step 10.8.2

Subtract

$π$ from

$4 π$ .

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

Step 10.9

Multiply

$2 π (0)$ .

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Step 10.9.1

Multiply

$0$ by

$2$ .

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

Step 10.9.2

Multiply

$0$ by

$π$ .

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

Step 10.10

Add

$\frac{3 π}{4}$ and

$0$ .

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

Step 10.11

Multiply the numerator by the reciprocal of the denominator.

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

Step 10.12

Cancel the common factor of

$3$ .

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Step 10.12.1

Factor

$3$ out of

$3 π$ .

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

Step 10.12.2

Cancel the common factor.

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

Step 10.12.3

Rewrite the expression.

$k = 0 : 3 c i s (\frac{π}{4})$

Step 11

Substitute

$k = 1$ into the formula and simplify.

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Step 11.1

Rewrite

$27$ as

$3^{3}$ .

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

Step 11.2

Apply the power rule and multiply exponents,

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

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

Step 11.3

Cancel the common factor of

$3$ .

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Step 11.3.1

Cancel the common factor.

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

Step 11.3.2

Rewrite the expression.

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

Step 11.4

Evaluate the exponent.

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

Step 11.5

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 11.6

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 11.7

Combine the numerators over the common denominator.

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

Step 11.8

Simplify the numerator.

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Step 11.8.1

Move

$4$ to the left of

$π$ .

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

Step 11.8.2

Subtract

$π$ from

$4 π$ .

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

Step 11.9

Multiply

$2$ by

$1$ .

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

Step 11.10

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 11.11

Combine

$2 π$ and

$\frac{4}{4}$ .

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

Step 11.12

Combine the numerators over the common denominator.

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

Step 11.13

Simplify the numerator.

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Step 11.13.1

Multiply

$4$ by

$2$ .

$k = 1 : 3 c i s (\frac{\frac{3 π + 8 π}{4}}{3})$

Step 11.13.2

Add

$3 π$ and

$8 π$ .

$k = 1 : 3 c i s (\frac{\frac{11 π}{4}}{3})$

Step 11.14

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 3 c i s (\frac{11 π}{4} \cdot \frac{1}{3})$

Step 11.15

Multiply

$\frac{11 π}{4} \cdot \frac{1}{3}$ .

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Step 11.15.1

Multiply

$\frac{11 π}{4}$ by

$\frac{1}{3}$ .

$k = 1 : 3 c i s (\frac{11 π}{4 \cdot 3})$

Step 11.15.2

Multiply

$4$ by

$3$ .

$k = 1 : 3 c i s (\frac{11 π}{12})$

Step 12

Substitute

$k = 2$ into the formula and simplify.

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Step 12.1

Rewrite

$27$ as

$3^{3}$ .

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

Step 12.2

Apply the power rule and multiply exponents,

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

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

Step 12.3

Cancel the common factor of

$3$ .

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Step 12.3.1

Cancel the common factor.

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

Step 12.3.2

Rewrite the expression.

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

Step 12.4

Evaluate the exponent.

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

Step 12.5

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 12.6

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 12.7

Combine the numerators over the common denominator.

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

Step 12.8

Simplify the numerator.

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Step 12.8.1

Move

$4$ to the left of

$π$ .

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

Step 12.8.2

Subtract

$π$ from

$4 π$ .

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

Step 12.9

Multiply

$2$ by

$2$ .

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

Step 12.10

To write

$4 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 12.11

Combine

$4 π$ and

$\frac{4}{4}$ .

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

Step 12.12

Combine the numerators over the common denominator.

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

Step 12.13

Simplify the numerator.

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Step 12.13.1

Multiply

$4$ by

$4$ .

$k = 2 : 3 c i s (\frac{\frac{3 π + 16 π}{4}}{3})$

Step 12.13.2

Add

$3 π$ and

$16 π$ .

$k = 2 : 3 c i s (\frac{\frac{19 π}{4}}{3})$

Step 12.14

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : 3 c i s (\frac{19 π}{4} \cdot \frac{1}{3})$

Step 12.15

Multiply

$\frac{19 π}{4} \cdot \frac{1}{3}$ .

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Step 12.15.1

Multiply

$\frac{19 π}{4}$ by

$\frac{1}{3}$ .

$k = 2 : 3 c i s (\frac{19 π}{4 \cdot 3})$

Step 12.15.2

Multiply

$4$ by

$3$ .

$k = 2 : 3 c i s (\frac{19 π}{12})$

Step 13

List the solutions.

$k = 0 : 3 c i s (\frac{π}{4})$

$k = 1 : 3 c i s (\frac{11 π}{12})$

$k = 2 : 3 c i s (\frac{19 π}{12})$

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