Trigonometry Examples

$θ^{2} = (\frac{4 π^{2}}{G M}) r^{3}$

Step 1

Rewrite the equation as $\frac{4 π^{2}}{G M} \cdot r^{3} = θ^{2}$ .

$\frac{4 π^{2}}{G M} \cdot r^{3} = θ^{2}$

Step 2

Combine $\frac{4 π^{2}}{G M}$ and $r^{3}$ .

$\frac{4 π^{2} r^{3}}{G M} = θ^{2}$

Step 3

Multiply both sides by $G M$ .

$\frac{4 π^{2} r^{3}}{G M} (G M) = θ^{2} (G M)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $G M$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{4 π^{2} r^{3}}{G M} (G M) = θ^{2} (G M)$

Step 4.1.1.2

Rewrite the expression.

$4 π^{2} r^{3} = θ^{2} (G M)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Remove parentheses.

$4 π^{2} r^{3} = θ^{2} G M$

Step 5

Solve for $r$ .

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Step 5.1

Divide each term in $4 π^{2} r^{3} = θ^{2} G M$ by $4 π^{2}$ and simplify.

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Step 5.1.1

Divide each term in $4 π^{2} r^{3} = θ^{2} G M$ by $4 π^{2}$ .

$\frac{4 π^{2} r^{3}}{4 π^{2}} = \frac{θ^{2} G M}{4 π^{2}}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $4$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{4 π^{2} r^{3}}{4 π^{2}} = \frac{θ^{2} G M}{4 π^{2}}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{π^{2} r^{3}}{π^{2}} = \frac{θ^{2} G M}{4 π^{2}}$

Step 5.1.2.2

Cancel the common factor of $π^{2}$ .

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Step 5.1.2.2.1

Cancel the common factor.

$\frac{π^{2} r^{3}}{π^{2}} = \frac{θ^{2} G M}{4 π^{2}}$

Step 5.1.2.2.2

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

$r^{3} = \frac{θ^{2} G M}{4 π^{2}}$

Step 5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \sqrt[3]{\frac{θ^{2} G M}{4 π^{2}}}$

Step 5.3

Simplify $\sqrt[3]{\frac{θ^{2} G M}{4 π^{2}}}$ .

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Step 5.3.1

Rewrite $\sqrt[3]{\frac{θ^{2} G M}{4 π^{2}}}$ as $\frac{\sqrt[3]{θ^{2} G M}}{\sqrt[3]{4 π^{2}}}$ .

$r = \frac{\sqrt[3]{θ^{2} G M}}{\sqrt[3]{4 π^{2}}}$

Step 5.3.2

Multiply $\frac{\sqrt[3]{θ^{2} G M}}{\sqrt[3]{4 π^{2}}}$ by $\frac{{\sqrt[3]{4 π^{2}}}^{2}}{{\sqrt[3]{4 π^{2}}}^{2}}$ .

$r = \frac{\sqrt[3]{θ^{2} G M}}{\sqrt[3]{4 π^{2}}} \cdot \frac{{\sqrt[3]{4 π^{2}}}^{2}}{{\sqrt[3]{4 π^{2}}}^{2}}$

Step 5.3.3

Combine and simplify the denominator.

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Step 5.3.3.1

Multiply $\frac{\sqrt[3]{θ^{2} G M}}{\sqrt[3]{4 π^{2}}}$ by $\frac{{\sqrt[3]{4 π^{2}}}^{2}}{{\sqrt[3]{4 π^{2}}}^{2}}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{\sqrt[3]{4 π^{2}} {\sqrt[3]{4 π^{2}}}^{2}}$

Step 5.3.3.2

Raise $\sqrt[3]{4 π^{2}}$ to the power of $1$ .

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{\sqrt[3]{4 π^{2}}}^{1} {\sqrt[3]{4 π^{2}}}^{2}}$

Step 5.3.3.3

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

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{\sqrt[3]{4 π^{2}}}^{1 + 2}}$

Step 5.3.3.4

Add $1$ and $2$ .

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{\sqrt[3]{4 π^{2}}}^{3}}$

Step 5.3.3.5

Rewrite ${\sqrt[3]{4 π^{2}}}^{3}$ as $4 π^{2}$ .

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Step 5.3.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 π^{2}}$ as ${(4 π^{2})}^{\frac{1}{3}}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{({(4 π^{2})}^{\frac{1}{3}})}^{3}}$

Step 5.3.3.5.2

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

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{(4 π^{2})}^{\frac{1}{3} \cdot 3}}$

Step 5.3.3.5.3

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

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{(4 π^{2})}^{\frac{3}{3}}}$

Step 5.3.3.5.4

Cancel the common factor of $3$ .

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Step 5.3.3.5.4.1

Cancel the common factor.

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{(4 π^{2})}^{\frac{3}{3}}}$

Step 5.3.3.5.4.2

Rewrite the expression.

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{{(4 π^{2})}^{1}}$

Step 5.3.3.5.5

Simplify.

$r = \frac{\sqrt[3]{θ^{2} G M} {\sqrt[3]{4 π^{2}}}^{2}}{4 π^{2}}$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Rewrite ${\sqrt[3]{4 π^{2}}}^{2}$ as $\sqrt[3]{{(4 π^{2})}^{2}}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{{(4 π^{2})}^{2}}}{4 π^{2}}$

Step 5.3.4.2

Apply the product rule to $4 π^{2}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{4^{2} {(π^{2})}^{2}}}{4 π^{2}}$

Step 5.3.4.3

Raise $4$ to the power of $2$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{16 {(π^{2})}^{2}}}{4 π^{2}}$

Step 5.3.4.4

Multiply the exponents in ${(π^{2})}^{2}$ .

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Step 5.3.4.4.1

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

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{16 π^{2 \cdot 2}}}{4 π^{2}}$

Step 5.3.4.4.2

Multiply $2$ by $2$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{16 π^{4}}}{4 π^{2}}$

Step 5.3.4.5

Rewrite $16 π^{4}$ as ${(2 π)}^{3} \cdot (2 π)$ .

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Step 5.3.4.5.1

Factor $8$ out of $16$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{8 (2) π^{4}}}{4 π^{2}}$

Step 5.3.4.5.2

Rewrite $8$ as $2^{3}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{2^{3} \cdot 2 π^{4}}}{4 π^{2}}$

Step 5.3.4.5.3

Factor out $π^{3}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{2^{3} \cdot 2 (π^{3} π)}}{4 π^{2}}$

Step 5.3.4.5.4

Move $2$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{2^{3} π^{3} \cdot 2 π}}{4 π^{2}}$

Step 5.3.4.5.5

Rewrite $2^{3} π^{3}$ as ${(2 π)}^{3}$ .

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{{(2 π)}^{3} \cdot 2 π}}{4 π^{2}}$

Step 5.3.4.5.6

Add parentheses.

$r = \frac{\sqrt[3]{θ^{2} G M} \sqrt[3]{{(2 π)}^{3} \cdot (2 π)}}{4 π^{2}}$

Step 5.3.4.6

Pull terms out from under the radical.

$r = \frac{\sqrt[3]{θ^{2} G M} \cdot 2 π \sqrt[3]{2 π}}{4 π^{2}}$

Step 5.3.4.7

Combine using the product rule for radicals.

$r = \frac{2 π \sqrt[3]{θ^{2} G M \cdot 2 π}}{4 π^{2}}$

Step 5.3.5

Reduce the expression by cancelling the common factors.

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Step 5.3.5.1

Cancel the common factor of $2$ and $4$ .

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Step 5.3.5.1.1

Factor $2$ out of $2 π \sqrt[3]{θ^{2} G M \cdot 2 π}$ .

$r = \frac{2 (π \sqrt[3]{θ^{2} G M \cdot 2 π})}{4 π^{2}}$

Step 5.3.5.1.2

Cancel the common factors.

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Step 5.3.5.1.2.1

Factor $2$ out of $4 π^{2}$ .

$r = \frac{2 (π \sqrt[3]{θ^{2} G M \cdot 2 π})}{2 (2 π^{2})}$

Step 5.3.5.1.2.2

Cancel the common factor.

$r = \frac{2 (π \sqrt[3]{θ^{2} G M \cdot 2 π})}{2 (2 π^{2})}$

Step 5.3.5.1.2.3

Rewrite the expression.

$r = \frac{π \sqrt[3]{θ^{2} G M \cdot 2 π}}{2 π^{2}}$

Step 5.3.5.2

Cancel the common factor of $π$ and $π^{2}$ .

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Step 5.3.5.2.1

Factor $π$ out of $π \sqrt[3]{θ^{2} G M \cdot 2 π}$ .

$r = \frac{π (\sqrt[3]{θ^{2} G M \cdot 2 π})}{2 π^{2}}$

Step 5.3.5.2.2

Cancel the common factors.

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Step 5.3.5.2.2.1

Factor $π$ out of $2 π^{2}$ .

$r = \frac{π (\sqrt[3]{θ^{2} G M \cdot 2 π})}{π (2 π)}$

Step 5.3.5.2.2.2

Cancel the common factor.

$r = \frac{π \sqrt[3]{θ^{2} G M \cdot 2 π}}{π (2 π)}$

Step 5.3.5.2.2.3

Rewrite the expression.

$r = \frac{\sqrt[3]{θ^{2} G M \cdot 2 π}}{2 π}$

Step 5.3.5.3

Reorder factors in $\frac{\sqrt[3]{θ^{2} G M \cdot 2 π}}{2 π}$ .

$r = \frac{\sqrt[3]{2 θ^{2} G M π}}{2 π}$