Trigonometry Examples

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

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

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} ⎞ ⎠

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

⎛ ⎝ \sqrt{1^{2} - {(\frac{2}{x})}^{2}}, 0 ⎞ ⎠

$(\sqrt{1^{2} - {(\frac{2}{x})}^{2}}, 0)$ , and the origin. Then

arcsin (\frac{2}{x})

$\arcsin (\frac{2}{x})$ 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} ⎞ ⎠

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

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

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

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

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

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

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

Step 1.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = \frac{2}{x}

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

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

$1$ as a fraction with a common denominator.

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

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

Step 3.2

Combine the numerators over the common denominator.

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

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

Step 3.3

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 3.4

Combine the numerators over the common denominator.

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

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

Step 3.5

Multiply

\frac{x + 2}{x}

$\frac{x + 2}{x}$ by

\frac{x - 2}{x}

$\frac{x - 2}{x}$ .

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

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

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