Precalculus Examples

$\cos (\arcsin (\frac{1}{2} x))$

Step 1

Write the expression using exponents.

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Step 1.1

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

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

Step 1.2

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

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

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 $b = \frac{1}{2} x$ .

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

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 3.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

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

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

Step 9

Multiply $2$ by $2$ .

$\sqrt{\frac{(2 + x) (2 - x)}{4}}$

Step 10

Rewrite $\frac{(2 + x) (2 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((2 + x) (2 - x))$ .

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Step 10.1

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

$\sqrt{\frac{1^{2} ((2 + x) (2 - x))}{4}}$

Step 10.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 10.3

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

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

Step 11

Pull terms out from under the radical.

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

Step 12

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

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