Evaluate cos(arcsin(( square root of 3)/2))

$\cos (\arcsin (\frac{\sqrt{3}}{2}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{\sqrt{3}}{2})}^{2}}, \frac{\sqrt{3}}{2})$ , $(\sqrt{1^{2} - {(\frac{\sqrt{3}}{2})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{\sqrt{3}}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{\sqrt{3}}{2})}^{2}}, \frac{\sqrt{3}}{2})$ . Therefore, $\cos (\arcsin (\frac{\sqrt{3}}{2}))$ is $\sqrt{\frac{1}{4}}$ .

$\sqrt{\frac{1}{4}}$

Step 2

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{\sqrt{1}}{\sqrt{4}}$

Step 3

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{4}}$

Step 4

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{1}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Evaluate cos(arcsin(( square root of 3)/2))

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