Trigonometry Examples

$\cos (arccsc (\frac{x}{2}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine into one fraction.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Multiply $- 1$ by $2$ .

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

Step 10

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

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

Step 11

Multiply $2$ by $2$ .

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 13

Pull terms out from under the radical.

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

Step 14

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

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

Step 15

Cancel the common factor of $2$ .

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Step 15.1

Cancel the common factor.

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

Step 15.2

Rewrite the expression.

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

Step 16

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

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