Simplify cos(arcsin(5/13))

$\cos (\arcsin (\frac{5}{13}))$

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{5}{13})}^{2}}, \frac{5}{13})$ , $(\sqrt{1^{2} - {(\frac{5}{13})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{5}{13})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{5}{13})}^{2}}, \frac{5}{13})$ . Therefore, $\cos (\arcsin (\frac{5}{13}))$ is $\frac{\sqrt{1^{2} - {(\frac{5}{13})}^{2}}}{1}$ .

$\frac{\sqrt{1^{2} - {(\frac{5}{13})}^{2}}}{1}$

Divide $\sqrt{1^{2} - {(\frac{5}{13})}^{2}}$ by $1$ .

$\sqrt{1^{2} - {(\frac{5}{13})}^{2}}$

One to any power is one.

$\sqrt{1 - {(\frac{5}{13})}^{2}}$

Apply the product rule to $\frac{5}{13}$ .

$\sqrt{1 - \frac{5^{2}}{13^{2}}}$

Raise $5$ to the power of $2$ .

$\sqrt{1 - \frac{25}{13^{2}}}$

Raise $13$ to the power of $2$ .

$\sqrt{1 - \frac{25}{169}}$

To write $\frac{1}{1}$ as a fraction with a common denominator, multiply by $\frac{169}{169}$ .

$\sqrt{\frac{1}{1} \cdot \frac{169}{169} - \frac{25}{169}}$

Write each expression with a common denominator of $169$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\sqrt{\frac{1 \cdot 169}{1 \cdot 169} - \frac{25}{169}}$

Multiply $169$ by $1$ .

$\sqrt{\frac{1 \cdot 169}{169} - \frac{25}{169}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{1 \cdot 169 - 25}{169}}$

Simplify the numerator.

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Multiply $1$ by $169$ .

$\sqrt{\frac{169 - 25}{169}}$

Subtract $25$ from $169$ .

$\sqrt{\frac{144}{169}}$

Rewrite $\sqrt{\frac{144}{169}}$ as $\frac{\sqrt{144}}{\sqrt{169}}$ .

$\frac{\sqrt{144}}{\sqrt{169}}$

Simplify the numerator.

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Rewrite $144$ as $12^{2}$ .

$\frac{\sqrt{12^{2}}}{\sqrt{169}}$

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

$\frac{12}{\sqrt{169}}$

Simplify the denominator.

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Rewrite $169$ as $13^{2}$ .

$\frac{12}{\sqrt{13^{2}}}$

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

$\frac{12}{13}$

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{13}$

Decimal Form:

$0.\overline{923076}$

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