Precalculus Examples

$\sin (arcsec (\frac{4}{x}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{4}{x}$ and $b = 1$ .

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

Step 5

Combine into one fraction.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\sqrt{\frac{4 + x}{x} (\frac{4}{x} - 1 \cdot \frac{x}{x})} \frac{x}{4}$

Step 7

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 8

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + x}{x} \cdot \frac{4 - x}{x}} \frac{x}{4}$

Step 9

Multiply $\frac{4 + x}{x}$ by $\frac{4 - x}{x}$ .

$\sqrt{\frac{(4 + x) (4 - x)}{x \cdot x}} \frac{x}{4}$

Step 10

Multiply $x$ by $x$ .

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

Step 11

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

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Step 11.1

Factor the perfect power $1^{2}$ out of $(4 + x) (4 - x)$ .

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

Pull terms out from under the radical.

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

Step 13

Combine $\frac{1}{x}$ and $\sqrt{(4 + x) (4 - x)}$ .

$\frac{\sqrt{(4 + x) (4 - x)}}{x} \cdot \frac{x}{4}$

Step 14

Cancel the common factor of $x$ .

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Step 14.1

Cancel the common factor.

$\frac{\sqrt{(4 + x) (4 - x)}}{x} \cdot \frac{x}{4}$

Step 14.2

Rewrite the expression.

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

Step 15

Combine $\sqrt{(4 + x) (4 - x)}$ and $\frac{1}{4}$ .

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