Precalculus Examples

$x = 4 + 2 \cos (t)$ , $y = - 1 + 4 \sin (t)$

Step 1

Remove parentheses.

$x = 4 + 2 \cos (\arcsin (\frac{y}{4} + \frac{1}{4}))$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2

Simplify $4 + 2 \cos (\arcsin (\frac{y}{4} + \frac{1}{4}))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{y}{4} + \frac{1}{4})}^{2}}, \frac{y}{4} + \frac{1}{4})$ , $(\sqrt{1^{2} - {(\frac{y}{4} + \frac{1}{4})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{y}{4} + \frac{1}{4})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{y}{4} + \frac{1}{4})}^{2}}, \frac{y}{4} + \frac{1}{4})$ . Therefore, $\cos (\arcsin (\frac{y}{4} + \frac{1}{4}))$ is $\sqrt{1 - {(\frac{y}{4} + \frac{1}{4})}^{2}}$ .

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

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.2

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

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

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.3

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 $b = \frac{y}{4} + \frac{1}{4}$ .

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

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

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

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4.2

Combine the numerators over the common denominator.

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

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4.3

Add $4$ and $1$ .

$x = 4 + 2 \sqrt{(\frac{y}{4} + \frac{5}{4}) (1 - (\frac{y}{4} + \frac{1}{4}))}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4.4

Apply the distributive property.

$x = 4 + 2 \sqrt{(\frac{y}{4} + \frac{5}{4}) (1 - \frac{y}{4} - \frac{1}{4})}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4.5

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

$x = 4 + 2 \sqrt{(\frac{y}{4} + \frac{5}{4}) (- \frac{y}{4} + \frac{4}{4} - \frac{1}{4})}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4.6

Combine the numerators over the common denominator.

$x = 4 + 2 \sqrt{(\frac{y}{4} + \frac{5}{4}) (- \frac{y}{4} + \frac{4 - 1}{4})}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.4.7

Subtract $1$ from $4$ .

$x = 4 + 2 \sqrt{(\frac{y}{4} + \frac{5}{4}) (- \frac{y}{4} + \frac{3}{4})}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

$x = 4 + 2 \sqrt{(\frac{y}{4} + \frac{5}{4}) (- \frac{y}{4} + \frac{3}{4})}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.5

Combine the numerators over the common denominator.

$x = 4 + 2 \sqrt{\frac{y + 5}{4} \cdot (- \frac{y}{4} + \frac{3}{4})}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.6

Combine the numerators over the common denominator.

$x = 4 + 2 \sqrt{\frac{y + 5}{4} \cdot \frac{- y + 3}{4}}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.7

Multiply $\frac{y + 5}{4}$ by $\frac{- y + 3}{4}$ .

$x = 4 + 2 \sqrt{\frac{(y + 5) (- y + 3)}{4 \cdot 4}}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.8

Multiply $4$ by $4$ .

$x = 4 + 2 \sqrt{\frac{(y + 5) (- y + 3)}{16}}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.9

Rewrite $\frac{(y + 5) (- y + 3)}{16}$ as ${(\frac{1}{4})}^{2} ((y + 5) (- y + 3))$ .

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Step 2.1.9.1

Factor the perfect power $1^{2}$ out of $(y + 5) (- y + 3)$ .

$x = 4 + 2 \sqrt{\frac{1^{2} ((y + 5) (- y + 3))}{16}}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.9.2

Factor the perfect power $4^{2}$ out of $16$ .

$x = 4 + 2 \sqrt{\frac{1^{2} ((y + 5) (- y + 3))}{4^{2} \cdot 1}}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.9.3

Rearrange the fraction $\frac{1^{2} ((y + 5) (- y + 3))}{4^{2} \cdot 1}$ .

$x = 4 + 2 \sqrt{{(\frac{1}{4})}^{2} ((y + 5) (- y + 3))}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

$x = 4 + 2 \sqrt{{(\frac{1}{4})}^{2} ((y + 5) (- y + 3))}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.10

Pull terms out from under the radical.

$x = 4 + 2 (\frac{1}{4} \cdot \sqrt{(y + 5) (- y + 3)})$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.11

Combine $\frac{1}{4}$ and $\sqrt{(y + 5) (- y + 3)}$ .

$x = 4 + 2 (\frac{\sqrt{(y + 5) (- y + 3)}}{4})$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.12

Cancel the common factor of $2$ .

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Step 2.1.12.1

Factor $2$ out of $4$ .

$x = 4 + 2 (\frac{\sqrt{(y + 5) (- y + 3)}}{2 (2)})$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.12.2

Cancel the common factor.

$x = 4 + 2 (\frac{\sqrt{(y + 5) (- y + 3)}}{2 \cdot 2})$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.1.12.3

Rewrite the expression.

$x = 4 + \frac{\sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

$x = 4 + \frac{\sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

$x = 4 + \frac{\sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 4 \cdot \frac{2}{2} + \frac{\sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.3

Combine $4$ and $\frac{2}{2}$ .

$x = \frac{4 \cdot 2}{2} + \frac{\sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.4

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 2 + \sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

Step 2.5

Multiply $4$ by $2$ .

$x = \frac{8 + \sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$

$x = \frac{8 + \sqrt{(y + 5) (- y + 3)}}{2}$

$t = \arcsin (\frac{y}{4} + \frac{1}{4})$