Trigonometry Examples

$\tan (\arccos (\frac{x}{3}))$

Step 1

Draw a triangle in the plane with vertices $(\frac{x}{3}, \sqrt{1^{2} - {(\frac{x}{3})}^{2}})$ , $(\frac{x}{3}, 0)$ , and the origin. Then $\arccos (\frac{x}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{x}{3}, \sqrt{1^{2} - {(\frac{x}{3})}^{2}})$ . Therefore, $\tan (\arccos (\frac{x}{3}))$ is $\frac{\sqrt{1 - {(\frac{x}{3})}^{2}}}{\frac{x}{3}}$ .

$\frac{\sqrt{1 - {(\frac{x}{3})}^{2}}}{\frac{x}{3}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1 - {(\frac{x}{3})}^{2}} \frac{3}{x}$

Step 3

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

$\sqrt{1^{2} - {(\frac{x}{3})}^{2}} \frac{3}{x}$

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 $b = \frac{x}{3}$ .

$\sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})} \frac{3}{x}$

Step 5

Simplify terms.

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Step 5.1

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

$\sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})} \frac{3}{x}$

Step 5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})} \frac{3}{x}$

Step 5.3

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

$\sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})} \frac{3}{x}$

Step 5.4

Combine the numerators over the common denominator.

$\sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}} \frac{3}{x}$

Step 5.5

Multiply $\frac{3 + x}{3}$ by $\frac{3 - x}{3}$ .

$\sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}} \frac{3}{x}$

Step 5.6

Multiply $3$ by $3$ .

$\sqrt{\frac{(3 + x) (3 - x)}{9}} \frac{3}{x}$

Step 6

Rewrite $\frac{(3 + x) (3 - x)}{9}$ as ${(\frac{1}{3})}^{2} ((3 + x) (3 - x))$ .

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Step 6.1

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

$\sqrt{\frac{1^{2} ((3 + x) (3 - x))}{9}} \frac{3}{x}$

Step 6.2

Factor the perfect power $3^{2}$ out of $9$ .

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

Step 6.3

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

$\sqrt{{(\frac{1}{3})}^{2} ((3 + x) (3 - x))} \frac{3}{x}$

Step 7

Pull terms out from under the radical.

$\frac{1}{3} \sqrt{(3 + x) (3 - x)} \frac{3}{x}$

Step 8

Combine $\frac{1}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$\frac{\sqrt{(3 + x) (3 - x)}}{3} \cdot \frac{3}{x}$

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Cancel the common factor.

$\frac{\sqrt{(3 + x) (3 - x)}}{3} \cdot \frac{3}{x}$

Step 9.2

Rewrite the expression.

$\sqrt{(3 + x) (3 - x)} \frac{1}{x}$

Step 10

Combine $\sqrt{(3 + x) (3 - x)}$ and $\frac{1}{x}$ .

$\frac{\sqrt{(3 + x) (3 - x)}}{x}$