Precalculus Examples

$x = 2 \sin (t)$ , $y = 2 \cos (t)$

Step 1

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

$x = 2 \sin (t)$

Step 2

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

$2 \sin (t) = x$

Step 3

Divide each term in $2 \sin (t) = x$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \sin (t) = x$ by $2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

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

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

Step 4

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

$t = \arcsin (\frac{x}{2})$

Step 5

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

$y = 2 \cos (\arcsin (\frac{x}{2}))$

Step 6

Remove parentheses.

$y = 2 \cos (\arcsin (\frac{x}{2}))$

Step 7

Simplify $2 \cos (\arcsin (\frac{x}{2}))$ .

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Step 7.1

Write the expression using exponents.

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Step 7.1.1

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

$y = 2 \sqrt{1 - {(\frac{x}{2})}^{2}}$

Step 7.1.2

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

$y = 2 \sqrt{1^{2} - {(\frac{x}{2})}^{2}}$

Step 7.2

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 $b = \frac{x}{2}$ .

$y = 2 \sqrt{(1 + \frac{x}{2}) (1 - \frac{x}{2})}$

Step 7.3

Simplify terms.

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Step 7.3.1

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

$y = 2 \sqrt{(\frac{2}{2} + \frac{x}{2}) (1 - \frac{x}{2})}$

Step 7.3.2

Combine the numerators over the common denominator.

$y = 2 \sqrt{\frac{2 + x}{2} (1 - \frac{x}{2})}$

Step 7.3.3

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

$y = 2 \sqrt{\frac{2 + x}{2} (\frac{2}{2} - \frac{x}{2})}$

Step 7.3.4

Combine the numerators over the common denominator.

$y = 2 \sqrt{\frac{2 + x}{2} \cdot \frac{2 - x}{2}}$

Step 7.3.5

Multiply $\frac{2 + x}{2}$ by $\frac{2 - x}{2}$ .

$y = 2 \sqrt{\frac{(2 + x) (2 - x)}{2 \cdot 2}}$

Step 7.3.6

Multiply $2$ by $2$ .

$y = 2 \sqrt{\frac{(2 + x) (2 - x)}{4}}$

Step 7.4

Rewrite $\frac{(2 + x) (2 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((2 + x) (2 - x))$ .

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Step 7.4.1

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

$y = 2 \sqrt{\frac{1^{2} ((2 + x) (2 - x))}{4}}$

Step 7.4.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 7.4.3

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

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

Step 7.5

Pull terms out from under the radical.

$y = 2 (\frac{1}{2} \sqrt{(2 + x) (2 - x)})$

Step 7.6

Combine $\frac{1}{2}$ and $\sqrt{(2 + x) (2 - x)}$ .

$y = 2 \frac{\sqrt{(2 + x) (2 - x)}}{2}$

Step 7.7

Cancel the common factor of $2$ .

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Step 7.7.1

Cancel the common factor.

$y = 2 \frac{\sqrt{(2 + x) (2 - x)}}{2}$

Step 7.7.2

Rewrite the expression.

$y = \sqrt{(2 + x) (2 - x)}$