Precalculus Examples

$x = 3 \sin^{3} (t)$

Step 1

Rewrite the equation as $3 \sin^{3} (t) = x$ .

$3 \sin^{3} (t) = x$

Step 2

Divide each term in $3 \sin^{3} (t) = x$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \sin^{3} (t) = x$ by $3$ .

$\frac{3 \sin^{3} (t)}{3} = \frac{x}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 \sin^{3} (t)}{3} = \frac{x}{3}$

Step 2.2.1.2

Divide $\sin^{3} (t)$ by $1$ .

$\sin^{3} (t) = \frac{x}{3}$

Step 3

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

$\sin (t) = \sqrt[3]{\frac{x}{3}}$

Step 4

Simplify $\sqrt[3]{\frac{x}{3}}$ .

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Step 4.1

Rewrite $\sqrt[3]{\frac{x}{3}}$ as $\frac{\sqrt[3]{x}}{\sqrt[3]{3}}$ .

$\sin (t) = \frac{\sqrt[3]{x}}{\sqrt[3]{3}}$

Step 4.2

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$\sin (t) = \frac{\sqrt[3]{x}}{\sqrt[3]{3}} \cdot \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}}$

Step 4.3.2

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

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}$

Step 4.3.3

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

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}}$

Step 4.3.4

Add $1$ and $2$ .

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{3}}$

Step 4.3.5

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

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Step 4.3.5.1

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

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{{(3^{\frac{1}{3}})}^{3}}$

Step 4.3.5.2

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

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}}$

Step 4.3.5.3

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

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 4.3.5.4

Cancel the common factor of $3$ .

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Step 4.3.5.4.1

Cancel the common factor.

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 4.3.5.4.2

Rewrite the expression.

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3^{1}}$

Step 4.3.5.5

Evaluate the exponent.

$\sin (t) = \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

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

$\sin (t) = \frac{\sqrt[3]{x} \sqrt[3]{3^{2}}}{3}$

Step 4.4.2

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

$\sin (t) = \frac{\sqrt[3]{x} \sqrt[3]{9}}{3}$

Step 4.5

Simplify with factoring out.

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Step 4.5.1

Combine using the product rule for radicals.

$\sin (t) = \frac{\sqrt[3]{x \cdot 9}}{3}$

Step 4.5.2

Reorder factors in $\frac{\sqrt[3]{x \cdot 9}}{3}$ .

$\sin (t) = \frac{\sqrt[3]{9 x}}{3}$

Step 5

Take the inverse sine of both sides of the equation to extract $t$ from inside the sine.

$t = \arcsin (\frac{\sqrt[3]{9 x}}{3})$