Trigonometry Examples

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

Step 1

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Add $2$ and $1$ .

$\sqrt{1 + {(- 1)}^{3} \frac{5^{2}}{13^{2}}}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Write $1$ as a fraction with a common denominator.

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Subtract $25$ from $169$ .

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

Step 10

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

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

Step 11

Simplify the numerator.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

Simplify the denominator.

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Step 12.1

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

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

Step 12.2

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

$\frac{12}{13}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{13}$

Decimal Form:

$0.\overline{923076}$