Find the Exact Value tan(arccos(-15/17))

$\tan (\arccos (- \frac{15}{17}))$

Draw a triangle in the plane with vertices $(- \frac{15}{17}, \sqrt{1^{2} - {(- \frac{15}{17})}^{2}})$ , $(- \frac{15}{17}, 0)$ , and the origin. Then $\arccos (- \frac{15}{17})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(- \frac{15}{17}, \sqrt{1^{2} - {(- \frac{15}{17})}^{2}})$ . Therefore, $\tan (\arccos (- \frac{15}{17}))$ is $\frac{\sqrt{1^{2} - {(- \frac{15}{17})}^{2}}}{- \frac{15}{17}}$ .

$\frac{\sqrt{1^{2} - {(- \frac{15}{17})}^{2}}}{- \frac{15}{17}}$

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1^{2} - {(- \frac{15}{17})}^{2}} (- \frac{17}{15})$

One to any power is one.

$\sqrt{1 - {(- \frac{15}{17})}^{2}} (- \frac{17}{15})$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{15}{17}$ .

$\sqrt{1 - ({(- 1)}^{2} {(\frac{15}{17})}^{2})} (- \frac{17}{15})$

Apply the product rule to $\frac{15}{17}$ .

$\sqrt{1 - ({(- 1)}^{2} \frac{15^{2}}{17^{2}})} (- \frac{17}{15})$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Move ${(- 1)}^{2}$ .

$\sqrt{1 + {(- 1)}^{2} \cdot - 1 \frac{15^{2}}{17^{2}}} (- \frac{17}{15})$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Raise $- 1$ to the power of $1$ .

$\sqrt{1 + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{15^{2}}{17^{2}}} (- \frac{17}{15})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{1 + {(- 1)}^{2 + 1} \frac{15^{2}}{17^{2}}} (- \frac{17}{15})$

Add $2$ and $1$ .

$\sqrt{1 + {(- 1)}^{3} \frac{15^{2}}{17^{2}}} (- \frac{17}{15})$

Raise $- 1$ to the power of $3$ .

$\sqrt{1 - \frac{15^{2}}{17^{2}}} (- \frac{17}{15})$

Raise $15$ to the power of $2$ .

$\sqrt{1 - \frac{225}{17^{2}}} (- \frac{17}{15})$

Raise $17$ to the power of $2$ .

$\sqrt{1 - \frac{225}{289}} (- \frac{17}{15})$

To write $\frac{1}{1}$ as a fraction with a common denominator, multiply by $\frac{289}{289}$ .

$\sqrt{\frac{1}{1} \cdot \frac{289}{289} - \frac{225}{289}} (- \frac{17}{15})$

Write each expression with a common denominator of $289$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\sqrt{\frac{1 \cdot 289}{1 \cdot 289} - \frac{225}{289}} (- \frac{17}{15})$

Multiply $289$ by $1$ .

$\sqrt{\frac{1 \cdot 289}{289} - \frac{225}{289}} (- \frac{17}{15})$

Combine the numerators over the common denominator.

$\sqrt{\frac{1 \cdot 289 - 225}{289}} (- \frac{17}{15})$

Simplify the numerator.

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Multiply $1$ by $289$ .

$\sqrt{\frac{289 - 225}{289}} (- \frac{17}{15})$

Subtract $225$ from $289$ .

$\sqrt{\frac{64}{289}} (- \frac{17}{15})$

Rewrite $\sqrt{\frac{64}{289}}$ as $\frac{\sqrt{64}}{\sqrt{289}}$ .

$\frac{\sqrt{64}}{\sqrt{289}} (- \frac{17}{15})$

Simplify the numerator.

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Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{8^{2}}}{\sqrt{289}} (- \frac{17}{15})$

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

$\frac{8}{\sqrt{289}} (- \frac{17}{15})$

Simplify the denominator.

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Rewrite $289$ as $17^{2}$ .

$\frac{8}{\sqrt{17^{2}}} (- \frac{17}{15})$

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

$\frac{8}{17} (- \frac{17}{15})$

Simplify terms.

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Cancel the common factor of $17$ .

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Move the leading negative in $- \frac{17}{15}$ into the numerator.

$\frac{8}{17} \cdot \frac{- 17}{15}$

Factor $17$ out of $- 17$ .

$\frac{8}{17} \cdot \frac{17 (- 1)}{15}$

Cancel the common factor.

$\frac{8}{17} \cdot \frac{17 \cdot - 1}{15}$

Rewrite the expression.

$8 (\frac{- 1}{15})$

Combine $8$ and $\frac{- 1}{15}$ .

$\frac{8 \cdot - 1}{15}$

Simplify the expression.

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Multiply $8$ by $- 1$ .

$\frac{- 8}{15}$

Move the negative in front of the fraction.

$- \frac{8}{15}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{15}$

Decimal Form:

$- 0.5\overline{3}$

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