Trigonometry Examples

$\frac{4}{3} π r^{3} = 36 π$

Step 1

Multiply both sides of the equation by $\frac{1}{\frac{4}{3} π}$ .

$\frac{1}{\frac{4}{3} π} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{1}{\frac{4}{3} π} (\frac{4}{3} \cdot (π r^{3}))$ .

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Step 2.1.1.1

Combine $\frac{4}{3}$ and $π$ .

$\frac{1}{\frac{4 π}{3}} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{4 π} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.3

Multiply $\frac{3}{4 π}$ by $1$ .

$\frac{3}{4 π} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.4

Cancel the common factor of $π$ .

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Step 2.1.1.4.1

Factor $π$ out of $4 π$ .

$\frac{3}{π \cdot 4} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.4.2

Factor $π$ out of $\frac{4}{3} \cdot (π r^{3})$ .

$\frac{3}{π \cdot 4} (π (\frac{4}{3} \cdot (r^{3}))) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.4.3

Cancel the common factor.

$\frac{3}{π \cdot 4} (π (\frac{4}{3} \cdot (r^{3}))) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.4.4

Rewrite the expression.

$\frac{3}{4} (\frac{4}{3} \cdot (r^{3})) = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.5

Combine $\frac{4}{3}$ and $r^{3}$ .

$\frac{3}{4} \cdot \frac{4 r^{3}}{3} = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.6

Multiply $\frac{3}{4}$ by $\frac{4 r^{3}}{3}$ .

$\frac{3 (4 r^{3})}{4 \cdot 3} = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.7

Multiply.

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Step 2.1.1.7.1

Multiply $4$ by $3$ .

$\frac{12 r^{3}}{4 \cdot 3} = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.7.2

Multiply $4$ by $3$ .

$\frac{12 r^{3}}{12} = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.8

Cancel the common factor of $12$ .

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Step 2.1.1.8.1

Cancel the common factor.

$\frac{12 r^{3}}{12} = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.1.1.8.2

Divide $r^{3}$ by $1$ .

$r^{3} = \frac{1}{\frac{4}{3} π} (36 π)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1}{\frac{4}{3} π} (36 π)$ .

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Step 2.2.1.1

Cancel the common factor of $π$ .

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Step 2.2.1.1.1

Factor $π$ out of $\frac{4}{3} π$ .

$r^{3} = \frac{1}{π \frac{4}{3}} (36 π)$

Step 2.2.1.1.2

Factor $π$ out of $36 π$ .

$r^{3} = \frac{1}{π \frac{4}{3}} (π \cdot 36)$

Step 2.2.1.1.3

Cancel the common factor.

$r^{3} = \frac{1}{π \frac{4}{3}} (π \cdot 36)$

Step 2.2.1.1.4

Rewrite the expression.

$r^{3} = \frac{1}{\frac{4}{3}} \cdot 36$

Step 2.2.1.2

Combine $\frac{1}{\frac{4}{3}}$ and $36$ .

$r^{3} = \frac{36}{\frac{4}{3}}$

Step 2.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$r^{3} = 36 (\frac{3}{4})$

Step 2.2.1.4

Cancel the common factor of $4$ .

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Step 2.2.1.4.1

Factor $4$ out of $36$ .

$r^{3} = 4 (9) \frac{3}{4}$

Step 2.2.1.4.2

Cancel the common factor.

$r^{3} = 4 \cdot 9 \frac{3}{4}$

Step 2.2.1.4.3

Rewrite the expression.

$r^{3} = 9 \cdot 3$

Step 2.2.1.5

Multiply $9$ by $3$ .

$r^{3} = 27$

Step 3

Subtract $27$ from both sides of the equation.

$r^{3} - 27 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Rewrite $27$ as $3^{3}$ .

$r^{3} - 3^{3} = 0$

Step 4.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = r$ and $b = 3$ .

$(r - 3) (r^{2} + r \cdot 3 + 3^{2}) = 0$

Step 4.3

Simplify.

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Step 4.3.1

Move $3$ to the left of $r$ .

$(r - 3) (r^{2} + 3 r + 3^{2}) = 0$

Step 4.3.2

Raise $3$ to the power of $2$ .

$(r - 3) (r^{2} + 3 r + 9) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r - 3 = 0$

$r^{2} + 3 r + 9 = 0$

Step 6

Set $r - 3$ equal to $0$ and solve for $r$ .

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Step 6.1

Set $r - 3$ equal to $0$ .

$r - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$r = 3$

Step 7

Set $r^{2} + 3 r + 9$ equal to $0$ and solve for $r$ .

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Step 7.1

Set $r^{2} + 3 r + 9$ equal to $0$ .

$r^{2} + 3 r + 9 = 0$

Step 7.2

Solve $r^{2} + 3 r + 9 = 0$ for $r$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $r$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

Raise $3$ to the power of $2$ .

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 7.2.3.1.2.2

Multiply $- 4$ by $9$ .

$r = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 7.2.3.1.3

Subtract $36$ from $9$ .

$r = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 7.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$r = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 7.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$r = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 7.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 7.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 7.2.3.1.7.1

Factor $9$ out of $27$ .

$r = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 7.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

$r = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 7.2.3.1.8

Pull terms out from under the radical.

$r = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 7.2.3.1.9

Move $3$ to the left of $i$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 7.2.3.2

Multiply $2$ by $1$ .

$r = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 7.2.4

The final answer is the combination of both solutions.

$r = - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 8

The final solution is all the values that make $(r - 3) (r^{2} + 3 r + 9) = 0$ true.

$r = 3, - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$