Trigonometry Examples

$\tan (\arccos (\frac{x}{2}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

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

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

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

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

Step 5.6

Multiply $2$ by $2$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

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

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

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

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