Precalculus Examples

$r = 3 + 3 \cos (θ)$

Step 1

Rewrite the equation as $3 + 3 \cos (θ) = r$ .

$3 + 3 \cos (θ) = r$

Step 2

Subtract $3$ from both sides of the equation.

$3 \cos (θ) = r - 3$

Step 3

Divide each term in $3 \cos (θ) = r - 3$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 \cos (θ) = r - 3$ by $3$ .

$\frac{3 \cos (θ)}{3} = \frac{r}{3} + \frac{- 3}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 \cos (θ)}{3} = \frac{r}{3} + \frac{- 3}{3}$

Step 3.2.1.2

Divide $\cos (θ)$ by $1$ .

$\cos (θ) = \frac{r}{3} + \frac{- 3}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 3$ by $3$ .

$\cos (θ) = \frac{r}{3} - 1$

Step 4

Take the inverse cosine of both sides of the equation to extract $θ$ from inside the cosine.

$θ = \arccos (\frac{r}{3} - 1)$

Step 5

To rewrite as a function of $r$ , write the equation so that $θ$ is by itself on one side of the equal sign and an expression involving only $r$ is on the other side.

$f (r) = \arccos (\frac{r}{3} - 1)$