Precalculus Examples

$x (t) = 7 - \sin (t)$ , $y (t) = 4 - 4 \sin (t)$

Step 1

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

$x = 7 - \sin (t)$

Step 2

Rewrite the equation as $7 - \sin (t) = x$ .

$7 - \sin (t) = x$

Step 3

Subtract $7$ from both sides of the equation.

$- \sin (t) = x - 7$

Step 4

Divide each term in $- \sin (t) = x - 7$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \sin (t) = x - 7$ by $- 1$ .

$\frac{- \sin (t)}{- 1} = \frac{x}{- 1} + \frac{- 7}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (t)}{1} = \frac{x}{- 1} + \frac{- 7}{- 1}$

Step 4.2.2

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

$\sin (t) = \frac{x}{- 1} + \frac{- 7}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$\sin (t) = - 1 \cdot x + \frac{- 7}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

$\sin (t) = - x + \frac{- 7}{- 1}$

Step 4.3.1.3

Divide $- 7$ by $- 1$ .

$\sin (t) = - x + 7$

Step 5

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

$t = \arcsin (- x + 7)$

Step 6

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

$y = 4 - 4 \sin (\arcsin (- x + 7))$

Step 7

Remove parentheses.

$y = 4 - 4 \sin (\arcsin (- x + 7))$

Step 8

Simplify $4 - 4 \sin (\arcsin (- x + 7))$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

The functions sine and arcsine are inverses.

$y = 4 - 4 (- x + 7)$

Step 8.1.2

Apply the distributive property.

$y = 4 - 4 (- x) - 4 \cdot 7$

Step 8.1.3

Multiply $- 1$ by $- 4$ .

$y = 4 + 4 x - 4 \cdot 7$

Step 8.1.4

Multiply $- 4$ by $7$ .

$y = 4 + 4 x - 28$

Step 8.2

Subtract $28$ from $4$ .

$y = 4 x - 24$