Precalculus Examples

$\tan (\arccos (\frac{x}{5}))$

Step 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, $\tan (\arccos (\frac{x}{5}))$ is $\frac{\sqrt{1 - {(\frac{x}{5})}^{2}}}{\frac{x}{5}}$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{5}$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

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

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

Step 5.6

Multiply $5$ by $5$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

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

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

Step 9

Cancel the common factor of $5$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

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

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