Precalculus Examples

$\cos (\arcsin (\frac{1}{9} x))$

Step 1

Write the expression using exponents.

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Step 1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{1}{9} x)}^{2}}, \frac{1}{9} x)$ , $(\sqrt{1^{2} - {(\frac{1}{9} x)}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{1}{9} x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{1}{9} x)}^{2}}, \frac{1}{9} x)$ . Therefore, $\cos (\arcsin (\frac{1}{9} x))$ is $\sqrt{1 - {(\frac{1}{9} x)}^{2}}$ .

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

Step 1.2

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

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

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 $b = \frac{1}{9} x$ .

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

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{9}$ and $x$ .

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

Step 3.2

Combine $\frac{1}{9}$ and $x$ .

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + x}{9} \cdot \frac{9 - x}{9}}$

Step 8

Multiply $\frac{9 + x}{9}$ by $\frac{9 - x}{9}$ .

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

Step 9

Multiply $9$ by $9$ .

$\sqrt{\frac{(9 + x) (9 - x)}{81}}$

Step 10

Rewrite $\frac{(9 + x) (9 - x)}{81}$ as ${(\frac{1}{9})}^{2} ((9 + x) (9 - x))$ .

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Step 10.1

Factor the perfect power $1^{2}$ out of $(9 + x) (9 - x)$ .

$\sqrt{\frac{1^{2} ((9 + x) (9 - x))}{81}}$

Step 10.2

Factor the perfect power $9^{2}$ out of $81$ .

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

Step 10.3

Rearrange the fraction $\frac{1^{2} ((9 + x) (9 - x))}{9^{2} \cdot 1}$ .

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

Step 11

Pull terms out from under the radical.

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

Step 12

Combine $\frac{1}{9}$ and $\sqrt{(9 + x) (9 - x)}$ .

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