Calculus Examples

$x (t) = 5 \cos (t)$ , $y (t) = 5 \sin (t)$

Step 1

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

$x = 5 \cos (t)$

Step 2

Rewrite the equation as $5 \cos (t) = x$ .

$5 \cos (t) = x$

Step 3

Divide each term in $5 \cos (t) = x$ by $5$ and simplify.

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Step 3.1

Divide each term in $5 \cos (t) = x$ by $5$ .

$\frac{5 \cos (t)}{5} = \frac{x}{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5 \cos (t)}{5} = \frac{x}{5}$

Step 3.2.1.2

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

$\cos (t) = \frac{x}{5}$

Step 4

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

$t = \arccos (\frac{x}{5})$

Step 5

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

$y = 5 \sin (\arccos (\frac{x}{5}))$

Step 6

Remove parentheses.

$y = 5 \sin (\arccos (\frac{x}{5}))$

Step 7

Simplify $5 \sin (\arccos (\frac{x}{5}))$ .

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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 $(\frac{x}{5}, \sqrt{1^{2} - {(\frac{x}{5})}^{2}})$ , $(\frac{x}{5}, 0)$ , and the origin. Then $\arccos (\frac{x}{5})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{x}{5}, \sqrt{1^{2} - {(\frac{x}{5})}^{2}})$ . Therefore, $\sin (\arccos (\frac{x}{5}))$ is $\sqrt{1 - {(\frac{x}{5})}^{2}}$ .

$y = 5 \sqrt{1 - {(\frac{x}{5})}^{2}}$

Step 7.1.2

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

$y = 5 \sqrt{1^{2} - {(\frac{x}{5})}^{2}}$

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 $b = \frac{x}{5}$ .

$y = 5 \sqrt{(1 + \frac{x}{5}) (1 - \frac{x}{5})}$

Step 7.3

Simplify terms.

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Step 7.3.1

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

$y = 5 \sqrt{(\frac{5}{5} + \frac{x}{5}) (1 - \frac{x}{5})}$

Step 7.3.2

Combine the numerators over the common denominator.

$y = 5 \sqrt{\frac{5 + x}{5} (1 - \frac{x}{5})}$

Step 7.3.3

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

$y = 5 \sqrt{\frac{5 + x}{5} (\frac{5}{5} - \frac{x}{5})}$

Step 7.3.4

Combine the numerators over the common denominator.

$y = 5 \sqrt{\frac{5 + x}{5} \cdot \frac{5 - x}{5}}$

Step 7.3.5

Multiply $\frac{5 + x}{5}$ by $\frac{5 - x}{5}$ .

$y = 5 \sqrt{\frac{(5 + x) (5 - x)}{5 \cdot 5}}$

Step 7.3.6

Multiply $5$ by $5$ .

$y = 5 \sqrt{\frac{(5 + x) (5 - x)}{25}}$

Step 7.4

Rewrite $\frac{(5 + x) (5 - x)}{25}$ as ${(\frac{1}{5})}^{2} ((5 + x) (5 - x))$ .

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Step 7.4.1

Factor the perfect power $1^{2}$ out of $(5 + x) (5 - x)$ .

$y = 5 \sqrt{\frac{1^{2} ((5 + x) (5 - x))}{25}}$

Step 7.4.2

Factor the perfect power $5^{2}$ out of $25$ .

$y = 5 \sqrt{\frac{1^{2} ((5 + x) (5 - x))}{5^{2} \cdot 1}}$

Step 7.4.3

Rearrange the fraction $\frac{1^{2} ((5 + x) (5 - x))}{5^{2} \cdot 1}$ .

$y = 5 \sqrt{{(\frac{1}{5})}^{2} ((5 + x) (5 - x))}$

Step 7.5

Pull terms out from under the radical.

$y = 5 (\frac{1}{5} \sqrt{(5 + x) (5 - x)})$

Step 7.6

Combine $\frac{1}{5}$ and $\sqrt{(5 + x) (5 - x)}$ .

$y = 5 \frac{\sqrt{(5 + x) (5 - x)}}{5}$

Step 7.7

Cancel the common factor of $5$ .

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Step 7.7.1

Cancel the common factor.

$y = 5 \frac{\sqrt{(5 + x) (5 - x)}}{5}$

Step 7.7.2

Rewrite the expression.

$y = \sqrt{(5 + x) (5 - x)}$