Trigonometry Examples

$\sqrt{\frac{1 - \frac{11}{23}}{1 + \frac{11}{23}}}$

Step 1

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

$\sqrt{\frac{\frac{23}{23} - \frac{11}{23}}{1 + \frac{11}{23}}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{23 - 11}{23}}{1 + \frac{11}{23}}}$

Step 3

Subtract $11$ from $23$ .

$\sqrt{\frac{\frac{12}{23}}{1 + \frac{11}{23}}}$

Step 4

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

$\sqrt{\frac{\frac{12}{23}}{\frac{23}{23} + \frac{11}{23}}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{12}{23}}{\frac{23 + 11}{23}}}$

Step 6

Add $23$ and $11$ .

$\sqrt{\frac{\frac{12}{23}}{\frac{34}{23}}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{12}{23} \cdot \frac{23}{34}}$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $12$ .

$\sqrt{\frac{2 (6)}{23} \cdot \frac{23}{34}}$

Step 8.2

Factor $2$ out of $34$ .

$\sqrt{\frac{2 \cdot 6}{23} \cdot \frac{23}{2 \cdot 17}}$

Step 8.3

Cancel the common factor.

$\sqrt{\frac{2 \cdot 6}{23} \cdot \frac{23}{2 \cdot 17}}$

Step 8.4

Rewrite the expression.

$\sqrt{\frac{6}{23} \cdot \frac{23}{17}}$

Step 9

Cancel the common factor of $23$ .

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Step 9.1

Cancel the common factor.

$\sqrt{\frac{6}{23} \cdot \frac{23}{17}}$

Step 9.2

Rewrite the expression.

$\sqrt{6 (\frac{1}{17})}$

Step 10

Combine $6$ and $\frac{1}{17}$ .

$\sqrt{\frac{6}{17}}$

Step 11

Rewrite $\sqrt{\frac{6}{17}}$ as $\frac{\sqrt{6}}{\sqrt{17}}$ .

$\frac{\sqrt{6}}{\sqrt{17}}$

Step 12

Multiply $\frac{\sqrt{6}}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{\sqrt{6}}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{\sqrt{6}}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{\sqrt{6} \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 13.2

Raise $\sqrt{17}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}}$

Step 13.3

Raise $\sqrt{17}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}}$

Step 13.4

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

$\frac{\sqrt{6} \sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Step 13.5

Add $1$ and $1$ .

$\frac{\sqrt{6} \sqrt{17}}{{\sqrt{17}}^{2}}$

Step 13.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{\sqrt{6} \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Step 13.6.2

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

$\frac{\sqrt{6} \sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Step 13.6.3

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

$\frac{\sqrt{6} \sqrt{17}}{17^{\frac{2}{2}}}$

Step 13.6.4

Cancel the common factor of $2$ .

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Step 13.6.4.1

Cancel the common factor.

$\frac{\sqrt{6} \sqrt{17}}{17^{\frac{2}{2}}}$

Step 13.6.4.2

Rewrite the expression.

$\frac{\sqrt{6} \sqrt{17}}{17^{1}}$

Step 13.6.5

Evaluate the exponent.

$\frac{\sqrt{6} \sqrt{17}}{17}$

Step 14

Simplify the numerator.

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Step 14.1

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 17}}{17}$

Step 14.2

Multiply $6$ by $17$ .

$\frac{\sqrt{102}}{17}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{102}}{17}$

Decimal Form:

$0.59408852 \dots$