Trigonometry Examples

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

Step 1

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

$- \sqrt{\frac{13}{13} - \frac{5}{13}}$

Step 2

Combine the numerators over the common denominator.

$- \sqrt{\frac{13 - 5}{13}}$

Step 3

Subtract $5$ from $13$ .

$- \sqrt{\frac{8}{13}}$

Step 4

Rewrite $\sqrt{\frac{8}{13}}$ as $\frac{\sqrt{8}}{\sqrt{13}}$ .

$- \frac{\sqrt{8}}{\sqrt{13}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.1

Factor $4$ out of $8$ .

$- \frac{\sqrt{4 (2)}}{\sqrt{13}}$

Step 5.1.2

Rewrite $4$ as $2^{2}$ .

$- \frac{\sqrt{2^{2} \cdot 2}}{\sqrt{13}}$

Step 5.2

Pull terms out from under the radical.

$- \frac{2 \sqrt{2}}{\sqrt{13}}$

Step 6

Multiply $\frac{2 \sqrt{2}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$- (\frac{2 \sqrt{2}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$- \frac{2 \sqrt{2} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 7.2

Raise $\sqrt{13}$ to the power of $1$ .

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

Step 7.3

Raise $\sqrt{13}$ to the power of $1$ .

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

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

Step 7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

$- \frac{2 \sqrt{2} \sqrt{13}}{13}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$- \frac{2 \sqrt{13 \cdot 2}}{13}$

Step 8.2

Multiply $13$ by $2$ .

$- \frac{2 \sqrt{26}}{13}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 \sqrt{26}}{13}$

Decimal Form:

$- 0.78446454 \dots$