Precalculus Examples

$\cos (\arcsin (\frac{x}{6}))$

Step 1

Write the expression using exponents.

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Step 1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{6})}^{2}}, \frac{x}{6})$ , $(\sqrt{1^{2} - {(\frac{x}{6})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{6})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{6})}^{2}}, \frac{x}{6})$ . Therefore, $\cos (\arcsin (\frac{x}{6}))$ is $\sqrt{1 - {(\frac{x}{6})}^{2}}$ .

$\sqrt{1 - {(\frac{x}{6})}^{2}}$

Step 1.2

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

$\sqrt{1^{2} - {(\frac{x}{6})}^{2}}$

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 $b = \frac{x}{6}$ .

$\sqrt{(1 + \frac{x}{6}) (1 - \frac{x}{6})}$

Step 3

Simplify terms.

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Step 3.1

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

$\sqrt{(\frac{6}{6} + \frac{x}{6}) (1 - \frac{x}{6})}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{6 + x}{6} (1 - \frac{x}{6})}$

Step 3.3

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

$\sqrt{\frac{6 + x}{6} (\frac{6}{6} - \frac{x}{6})}$

Step 3.4

Combine the numerators over the common denominator.

$\sqrt{\frac{6 + x}{6} \cdot \frac{6 - x}{6}}$

Step 3.5

Multiply $\frac{6 + x}{6}$ by $\frac{6 - x}{6}$ .

$\sqrt{\frac{(6 + x) (6 - x)}{6 \cdot 6}}$

Step 3.6

Multiply $6$ by $6$ .

$\sqrt{\frac{(6 + x) (6 - x)}{36}}$

Step 4

Rewrite $\frac{(6 + x) (6 - x)}{36}$ as ${(\frac{1}{6})}^{2} ((6 + x) (6 - x))$ .

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Step 4.1

Factor the perfect power $1^{2}$ out of $(6 + x) (6 - x)$ .

$\sqrt{\frac{1^{2} ((6 + x) (6 - x))}{36}}$

Step 4.2

Factor the perfect power $6^{2}$ out of $36$ .

$\sqrt{\frac{1^{2} ((6 + x) (6 - x))}{6^{2} \cdot 1}}$

Step 4.3

Rearrange the fraction $\frac{1^{2} ((6 + x) (6 - x))}{6^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{6})}^{2} ((6 + x) (6 - x))}$

Step 5

Pull terms out from under the radical.

$\frac{1}{6} \sqrt{(6 + x) (6 - x)}$

Step 6

Combine $\frac{1}{6}$ and $\sqrt{(6 + x) (6 - x)}$ .

$\frac{\sqrt{(6 + x) (6 - x)}}{6}$