Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

$\frac{12}{13}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{13}$

Decimal Form:

$0.\overline{923076}$