Precalculus Examples

$\cos (\arcsin (\frac{1}{7} x))$

Step 1

Write the expression using exponents.

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Step 1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{1}{7} x)}^{2}}, \frac{1}{7} x)$ , $(\sqrt{1^{2} - {(\frac{1}{7} x)}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{1}{7} x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{1}{7} x)}^{2}}, \frac{1}{7} x)$ . Therefore, $\cos (\arcsin (\frac{1}{7} x))$ is $\sqrt{1 - {(\frac{1}{7} x)}^{2}}$ .

$\sqrt{1 - {(\frac{1}{7} x)}^{2}}$

Step 1.2

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

$\sqrt{1^{2} - {(\frac{1}{7} x)}^{2}}$

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 $b = \frac{1}{7} x$ .

$\sqrt{(1 + \frac{1}{7} x) (1 - (\frac{1}{7} x))}$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{7}$ and $x$ .

$\sqrt{(1 + \frac{x}{7}) (1 - (\frac{1}{7} x))}$

Step 3.2

Combine $\frac{1}{7}$ and $x$ .

$\sqrt{(1 + \frac{x}{7}) (1 - \frac{x}{7})}$

Step 4

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

$\sqrt{(\frac{7}{7} + \frac{x}{7}) (1 - \frac{x}{7})}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{7 + x}{7} (1 - \frac{x}{7})}$

Step 6

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

$\sqrt{\frac{7 + x}{7} (\frac{7}{7} - \frac{x}{7})}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{7 + x}{7} \cdot \frac{7 - x}{7}}$

Step 8

Multiply $\frac{7 + x}{7}$ by $\frac{7 - x}{7}$ .

$\sqrt{\frac{(7 + x) (7 - x)}{7 \cdot 7}}$

Step 9

Multiply $7$ by $7$ .

$\sqrt{\frac{(7 + x) (7 - x)}{49}}$

Step 10

Rewrite $\frac{(7 + x) (7 - x)}{49}$ as ${(\frac{1}{7})}^{2} ((7 + x) (7 - x))$ .

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Step 10.1

Factor the perfect power $1^{2}$ out of $(7 + x) (7 - x)$ .

$\sqrt{\frac{1^{2} ((7 + x) (7 - x))}{49}}$

Step 10.2

Factor the perfect power $7^{2}$ out of $49$ .

$\sqrt{\frac{1^{2} ((7 + x) (7 - x))}{7^{2} \cdot 1}}$

Step 10.3

Rearrange the fraction $\frac{1^{2} ((7 + x) (7 - x))}{7^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{7})}^{2} ((7 + x) (7 - x))}$

Step 11

Pull terms out from under the radical.

$\frac{1}{7} \sqrt{(7 + x) (7 - x)}$

Step 12

Combine $\frac{1}{7}$ and $\sqrt{(7 + x) (7 - x)}$ .

$\frac{\sqrt{(7 + x) (7 - x)}}{7}$