Algebra Examples

$x = h + r \cos (t)$ , $y = k + r \sin (t)$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = h + r \cos (t)$

Step 2

Rewrite the equation as $h + r \cos (t) = x$ .

$h + r \cos (t) = x$

Step 3

Subtract $h$ from both sides of the equation.

$r \cos (t) = x - h$

Step 4

Divide each term in $r \cos (t) = x - h$ by $r$ and simplify.

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Step 4.1

Divide each term in $r \cos (t) = x - h$ by $r$ .

$\frac{r \cos (t)}{r} = \frac{x}{r} + \frac{- h}{r}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $r$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{r \cos (t)}{r} = \frac{x}{r} + \frac{- h}{r}$

Step 4.2.1.2

Divide $\cos (t)$ by $1$ .

$\cos (t) = \frac{x}{r} + \frac{- h}{r}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$\cos (t) = \frac{x}{r} - \frac{h}{r}$

Step 5

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

$t = \arccos (\frac{x}{r} - \frac{h}{r})$

Step 6

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = k + r \sin (\arccos (\frac{x}{r} - \frac{h}{r}))$

Step 7

Remove parentheses.

$y = k + r \sin (\arccos (\frac{x}{r} - \frac{h}{r}))$

Step 8

Simplify $k + r \sin (\arccos (\frac{x}{r} - \frac{h}{r}))$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Draw a triangle in the plane with vertices $(\frac{x}{r} - \frac{h}{r}, \sqrt{1^{2} - {(\frac{x}{r} - \frac{h}{r})}^{2}})$ , $(\frac{x}{r} - \frac{h}{r}, 0)$ , and the origin. Then $\arccos (\frac{x}{r} - \frac{h}{r})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{x}{r} - \frac{h}{r}, \sqrt{1^{2} - {(\frac{x}{r} - \frac{h}{r})}^{2}})$ . Therefore, $\sin (\arccos (\frac{x}{r} - \frac{h}{r}))$ is $\sqrt{1 - {(\frac{x}{r} - \frac{h}{r})}^{2}}$ .

$y = k + r \sqrt{1 - {(\frac{x}{r} - \frac{h}{r})}^{2}}$

Step 8.1.2

Rewrite $1$ as $1^{2}$ .

$y = k + r \sqrt{1^{2} - {(\frac{x}{r} - \frac{h}{r})}^{2}}$

Step 8.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{r} - \frac{h}{r}$ .

$y = k + r \sqrt{(1 + \frac{x}{r} - \frac{h}{r}) (1 - (\frac{x}{r} - \frac{h}{r}))}$

Step 8.1.4

Simplify.

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Step 8.1.4.1

Apply the distributive property.

$y = k + r \sqrt{(1 + \frac{x}{r} - \frac{h}{r}) (1 - \frac{x}{r} - - \frac{h}{r})}$

Step 8.1.4.2

Multiply $- - \frac{h}{r}$ .

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Step 8.1.4.2.1

Multiply $- 1$ by $- 1$ .

$y = k + r \sqrt{(1 + \frac{x}{r} - \frac{h}{r}) (1 - \frac{x}{r} + 1 \frac{h}{r})}$

Step 8.1.4.2.2

Multiply $\frac{h}{r}$ by $1$ .

$y = k + r \sqrt{(1 + \frac{x}{r} - \frac{h}{r}) (1 - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.5

Write $1$ as a fraction with a common denominator.

$y = k + r \sqrt{(\frac{r}{r} + \frac{x}{r} - \frac{h}{r}) (1 - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.6

Combine the numerators over the common denominator.

$y = k + r \sqrt{(\frac{r + x}{r} - \frac{h}{r}) (1 - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.7

Combine the numerators over the common denominator.

$y = k + r \sqrt{\frac{r + x - h}{r} (1 - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.8

Simplify the numerator.

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Step 8.1.8.1

Move $x$ .

$y = k + r \sqrt{\frac{r - h + x}{r} (1 - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.8.2

Reorder $r$ and $- h$ .

$y = k + r \sqrt{\frac{- h + r + x}{r} (1 - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.9

Write $1$ as a fraction with a common denominator.

$y = k + r \sqrt{\frac{- h + r + x}{r} (\frac{r}{r} - \frac{x}{r} + \frac{h}{r})}$

Step 8.1.10

Combine the numerators over the common denominator.

$y = k + r \sqrt{\frac{- h + r + x}{r} (\frac{r - x}{r} + \frac{h}{r})}$

Step 8.1.11

Combine the numerators over the common denominator.

$y = k + r \sqrt{\frac{- h + r + x}{r} \cdot \frac{r - x + h}{r}}$

Step 8.1.12

Simplify the numerator.

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Step 8.1.12.1

Move $- x$ .

$y = k + r \sqrt{\frac{- h + r + x}{r} \cdot \frac{r + h - x}{r}}$

Step 8.1.12.2

Reorder $r$ and $h$ .

$y = k + r \sqrt{\frac{- h + r + x}{r} \cdot \frac{h + r - x}{r}}$

Step 8.1.13

Multiply $\frac{- h + r + x}{r}$ by $\frac{h + r - x}{r}$ .

$y = k + r \sqrt{\frac{(- h + r + x) (h + r - x)}{r \cdot r}}$

Step 8.1.14

Multiply $r$ by $r$ .

$y = k + r \sqrt{\frac{(- h + r + x) (h + r - x)}{r^{2}}}$

Step 8.1.15

Rewrite $\frac{(- h + r + x) (h + r - x)}{r^{2}}$ as ${(\frac{1}{r})}^{2} ((- h + r + x) (h + r - x))$ .

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Step 8.1.15.1

Factor the perfect power $1^{2}$ out of $(- h + r + x) (h + r - x)$ .

$y = k + r \sqrt{\frac{1^{2} ((- h + r + x) (h + r - x))}{r^{2}}}$

Step 8.1.15.2

Factor the perfect power $r^{2}$ out of $r^{2}$ .

$y = k + r \sqrt{\frac{1^{2} ((- h + r + x) (h + r - x))}{r^{2} \cdot 1}}$

Step 8.1.15.3

Rearrange the fraction $\frac{1^{2} ((- h + r + x) (h + r - x))}{r^{2} \cdot 1}$ .

$y = k + r \sqrt{{(\frac{1}{r})}^{2} ((- h + r + x) (h + r - x))}$

Step 8.1.16

Pull terms out from under the radical.

$y = k + r (\frac{1}{r} \sqrt{(- h + r + x) (h + r - x)})$

Step 8.1.17

Combine $\frac{1}{r}$ and $\sqrt{(- h + r + x) (h + r - x)}$ .

$y = k + r \frac{\sqrt{(- h + r + x) (h + r - x)}}{r}$

Step 8.1.18

Cancel the common factor of $r$ .

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Step 8.1.18.1

Cancel the common factor.

$y = k + r \frac{\sqrt{(- h + r + x) (h + r - x)}}{r}$

Step 8.1.18.2

Rewrite the expression.

$y = k + \sqrt{(- h + r + x) (h + r - x)}$

Step 8.2

Reorder $k$ and $\sqrt{(- h + r + x) (h + r - x)}$ .

$y = \sqrt{(- h + r + x) (h + r - x)} + k$