Trigonometry Examples

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

Step 1

Write the expression using exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

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

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

Step 3.6

Multiply $x$ by $x$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

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