Find the Exact Value tan(arccos(12/13))

$\tan (\arccos (\frac{12}{13}))$

Draw a triangle in the plane with vertices $(\frac{12}{13}, \sqrt{1^{2} - {(\frac{12}{13})}^{2}})$ , $(\frac{12}{13}, 0)$ , and the origin. Then $\arccos (\frac{12}{13})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{12}{13}, \sqrt{1^{2} - {(\frac{12}{13})}^{2}})$ . Therefore, $\tan (\arccos (\frac{12}{13}))$ is $\frac{\sqrt{1^{2} - {(\frac{12}{13})}^{2}}}{\frac{12}{13}}$ .

$\frac{\sqrt{1^{2} - {(\frac{12}{13})}^{2}}}{\frac{12}{13}}$

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1^{2} - {(\frac{12}{13})}^{2}} \frac{13}{12}$

One to any power is one.

$\sqrt{1 - {(\frac{12}{13})}^{2}} \frac{13}{12}$

Apply the product rule to $\frac{12}{13}$ .

$\sqrt{1 - \frac{12^{2}}{13^{2}}} \frac{13}{12}$

Raise $12$ to the power of $2$ .

$\sqrt{1 - \frac{144}{13^{2}}} \frac{13}{12}$

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

$\sqrt{1 - \frac{144}{169}} \frac{13}{12}$

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

$\sqrt{\frac{1}{1} \cdot \frac{169}{169} - \frac{144}{169}} \frac{13}{12}$

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

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Combine.

$\sqrt{\frac{1 \cdot 169}{1 \cdot 169} - \frac{144}{169}} \frac{13}{12}$

Multiply $169$ by $1$ .

$\sqrt{\frac{1 \cdot 169}{169} - \frac{144}{169}} \frac{13}{12}$

Combine the numerators over the common denominator.

$\sqrt{\frac{1 \cdot 169 - 144}{169}} \frac{13}{12}$

Simplify the numerator.

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Multiply $1$ by $169$ .

$\sqrt{\frac{169 - 144}{169}} \frac{13}{12}$

Subtract $144$ from $169$ .

$\sqrt{\frac{25}{169}} \frac{13}{12}$

Rewrite $\sqrt{\frac{25}{169}}$ as $\frac{\sqrt{25}}{\sqrt{169}}$ .

$\frac{\sqrt{25}}{\sqrt{169}} \cdot \frac{13}{12}$

Simplify the numerator.

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Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{169}} \cdot \frac{13}{12}$

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

$\frac{5}{\sqrt{169}} \cdot \frac{13}{12}$

Simplify the denominator.

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

$\frac{5}{\sqrt{13^{2}}} \cdot \frac{13}{12}$

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

$\frac{5}{13} \cdot \frac{13}{12}$

Simplify terms.

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Cancel the common factor of $13$ .

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Cancel the common factor.

$\frac{5}{13} \cdot \frac{13}{12}$

Rewrite the expression.

$5 (\frac{1}{12})$

Combine $5$ and $\frac{1}{12}$ .

$\frac{5}{12}$

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{12}$

Decimal Form:

$0.41\overline{6}$

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