Find the Exact Value sin(arccos(x/2))

$\sin (\arccos (\frac{x}{2}))$

Write the expression using exponents.

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

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

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

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

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

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

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

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

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

Combine the numerators over the common denominator.

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Pull terms out from under the radical.

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

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

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

Find the Exact Value sin(arccos(x/2))

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