Precalculus Examples

$r^{2} \cos^{3} (θ) = \sin (θ)$

Step 1

Divide each term in $r^{2} \cos^{3} (θ) = \sin (θ)$ by $\cos^{3} (θ)$ and simplify.

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Step 1.1

Divide each term in $r^{2} \cos^{3} (θ) = \sin (θ)$ by $\cos^{3} (θ)$ .

$\frac{r^{2} \cos^{3} (θ)}{\cos^{3} (θ)} = \frac{\sin (θ)}{\cos^{3} (θ)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\cos^{3} (θ)$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{r^{2} \cos^{3} (θ)}{\cos^{3} (θ)} = \frac{\sin (θ)}{\cos^{3} (θ)}$

Step 1.2.1.2

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{\sin (θ)}{\cos^{3} (θ)}$

Step 2

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

$r = \pm \sqrt{\frac{\sin (θ)}{\cos^{3} (θ)}}$

Step 3

Simplify $\pm \sqrt{\frac{\sin (θ)}{\cos^{3} (θ)}}$ .

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Step 3.1

Rewrite $\frac{\sin (θ)}{\cos^{3} (θ)}$ as ${(\frac{1}{\cos (θ)})}^{2} \frac{\sin (θ)}{\cos (θ)}$ .

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Step 3.1.1

Factor the perfect power $1^{2}$ out of $\sin (θ)$ .

$r = \pm \sqrt{\frac{1^{2} \sin (θ)}{\cos^{3} (θ)}}$

Step 3.1.2

Factor the perfect power $\cos^{2} (θ)$ out of $\cos^{3} (θ)$ .

$r = \pm \sqrt{\frac{1^{2} \sin (θ)}{\cos^{2} (θ) \cos (θ)}}$

Step 3.1.3

Rearrange the fraction $\frac{1^{2} \sin (θ)}{\cos^{2} (θ) \cos (θ)}$ .

$r = \pm \sqrt{{(\frac{1}{\cos (θ)})}^{2} \frac{\sin (θ)}{\cos (θ)}}$

Step 3.2

Pull terms out from under the radical.

$r = \pm \frac{1}{\cos (θ)} \sqrt{\frac{\sin (θ)}{\cos (θ)}}$

Step 3.3

Rewrite $\sqrt{\frac{\sin (θ)}{\cos (θ)}}$ as $\frac{\sqrt{\sin (θ)}}{\sqrt{\cos (θ)}}$ .

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)}}{\sqrt{\cos (θ)}}$

Step 3.4

Multiply $\frac{\sqrt{\sin (θ)}}{\sqrt{\cos (θ)}}$ by $\frac{\sqrt{\cos (θ)}}{\sqrt{\cos (θ)}}$ .

$r = \pm \frac{1}{\cos (θ)} (\frac{\sqrt{\sin (θ)}}{\sqrt{\cos (θ)}} \cdot \frac{\sqrt{\cos (θ)}}{\sqrt{\cos (θ)}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{\sqrt{\sin (θ)}}{\sqrt{\cos (θ)}}$ by $\frac{\sqrt{\cos (θ)}}{\sqrt{\cos (θ)}}$ .

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{\sqrt{\cos (θ)} \sqrt{\cos (θ)}}$

Step 3.5.2

Raise $\sqrt{\cos (θ)}$ to the power of $1$ .

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\sqrt{\cos (θ)}}^{1} \sqrt{\cos (θ)}}$

Step 3.5.3

Raise $\sqrt{\cos (θ)}$ to the power of $1$ .

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\sqrt{\cos (θ)}}^{1} {\sqrt{\cos (θ)}}^{1}}$

Step 3.5.4

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

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\sqrt{\cos (θ)}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\sqrt{\cos (θ)}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{\cos (θ)}}^{2}$ as $\cos (θ)$ .

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Step 3.5.6.1

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

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{({\cos (θ)}^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\cos (θ)}^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\cos (θ)}^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{{\cos (θ)}^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{\cos^{1} (θ)}$

Step 3.5.6.5

Simplify.

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ)} \sqrt{\cos (θ)}}{\cos (θ)}$

Step 3.6

Combine using the product rule for radicals.

$r = \pm \frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos (θ)}$

Step 3.7

Multiply $\frac{1}{\cos (θ)} \cdot \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos (θ)}$ .

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Step 3.7.1

Multiply $\frac{1}{\cos (θ)}$ by $\frac{\sqrt{\sin (θ) \cos (θ)}}{\cos (θ)}$ .

$r = \pm \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos (θ) \cos (θ)}$

Step 3.7.2

Raise $\cos (θ)$ to the power of $1$ .

$r = \pm \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{1} (θ) \cos (θ)}$

Step 3.7.3

Raise $\cos (θ)$ to the power of $1$ .

$r = \pm \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{1} (θ) \cos^{1} (θ)}$

Step 3.7.4

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

$r = \pm \frac{\sqrt{\sin (θ) \cos (θ)}}{{\cos (θ)}^{1 + 1}}$

Step 3.7.5

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$

$r = - \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$

$r = \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$

$r = - \frac{\sqrt{\sin (θ) \cos (θ)}}{\cos^{2} (θ)}$