Calculus Examples

$x = 7 \cos (t)$ , $y = 3 \cos (7 t)$

Step 1

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

$x = 7 \cos (t)$

Step 2

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

$7 \cos (t) = x$

Step 3

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

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Step 3.1

Divide each term in $7 \cos (t) = x$ by $7$ .

$\frac{7 \cos (t)}{7} = \frac{x}{7}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{7 \cos (t)}{7} = \frac{x}{7}$

Step 3.2.1.2

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

$\cos (t) = \frac{x}{7}$

Step 4

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

$t = \arccos (\frac{x}{7})$

Step 5

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

$y = 3 \cos (7 (\arccos (\frac{x}{7})))$

Step 6

Multiply $7$ by $\arccos (\frac{x}{7})$ .

$y = 3 \cos (7 \arccos (\frac{x}{7}))$