Calculus Examples

$x = \sin^{2} (t)$ and $y = \cos (t)$

Step 1

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

$x = \sin^{2} (t)$

Step 2

Rewrite the equation as $\sin^{2} (t) = x$ .

$\sin^{2} (t) = x$

Step 3

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

$\sin (t) = \pm \sqrt{x}$

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.

$\sin (t) = \sqrt{x}$

Step 4.2

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

$\sin (t) = - \sqrt{x}$

Step 4.3

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

$\sin (t) = \sqrt{x}, - \sqrt{x}$

Step 5

Set up each of the solutions to solve for $t$ .

$\sin (t) = \sqrt{x}$

$\sin (t) = - \sqrt{x}$

Step 6

Solve for $t$ in $\sin (t) = \sqrt{x}$ .

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Step 6.1

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

$t = \arcsin (\sqrt{x})$

Step 7

Solve for $t$ in $\sin (t) = - \sqrt{x}$ .

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Step 7.1

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

$t = \arcsin (- \sqrt{x})$

Step 8

List all of the solutions.

$t = \arcsin (\sqrt{x})$

$t = \arcsin (- \sqrt{x})$

Step 9

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

$y = \cos ((\arcsin (\sqrt{x}), \arcsin (- \sqrt{x})))$