Trigonometry Examples

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

Step 1

Divide $\sqrt{11}$ by $1$ .

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

Step 2

Divide $\sqrt{11}$ by $1$ .

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

Step 3

Multiply $\frac{1 + \sqrt{11}}{1 - \sqrt{11}}$ by $\frac{1 + \sqrt{11}}{1 + \sqrt{11}}$ .

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

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{1 + \sqrt{11}}{1 - \sqrt{11}}$ by $\frac{1 + \sqrt{11}}{1 + \sqrt{11}}$ .

$\sqrt{\frac{(1 + \sqrt{11}) (1 + \sqrt{11})}{(1 - \sqrt{11}) (1 + \sqrt{11})}}$

Step 4.2

Expand the denominator using the FOIL method.

$\sqrt{\frac{(1 + \sqrt{11}) (1 + \sqrt{11})}{1 + \sqrt{11} - \sqrt{11} - {\sqrt{11}}^{2}}}$

Step 4.3

Simplify.

$\sqrt{\frac{(1 + \sqrt{11}) (1 + \sqrt{11})}{- 10}}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $1 + \sqrt{11}$ to the power of $1$ .

$\sqrt{\frac{{(1 + \sqrt{11})}^{1} (1 + \sqrt{11})}{- 10}}$

Step 5.2

Raise $1 + \sqrt{11}$ to the power of $1$ .

$\sqrt{\frac{{(1 + \sqrt{11})}^{1} {(1 + \sqrt{11})}^{1}}{- 10}}$

Step 5.3

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

$\sqrt{\frac{{(1 + \sqrt{11})}^{1 + 1}}{- 10}}$

Step 5.4

Add $1$ and $1$ .

$\sqrt{\frac{{(1 + \sqrt{11})}^{2}}{- 10}}$

Step 6

Rewrite ${(1 + \sqrt{11})}^{2}$ as $(1 + \sqrt{11}) (1 + \sqrt{11})$ .

$\sqrt{\frac{(1 + \sqrt{11}) (1 + \sqrt{11})}{- 10}}$

Step 7

Expand $(1 + \sqrt{11}) (1 + \sqrt{11})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\sqrt{\frac{1 (1 + \sqrt{11}) + \sqrt{11} (1 + \sqrt{11})}{- 10}}$

Step 7.2

Apply the distributive property.

$\sqrt{\frac{1 \cdot 1 + 1 \sqrt{11} + \sqrt{11} (1 + \sqrt{11})}{- 10}}$

Step 7.3

Apply the distributive property.

$\sqrt{\frac{1 \cdot 1 + 1 \sqrt{11} + \sqrt{11} \cdot 1 + \sqrt{11} \sqrt{11}}{- 10}}$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $1$ by $1$ .

$\sqrt{\frac{1 + 1 \sqrt{11} + \sqrt{11} \cdot 1 + \sqrt{11} \sqrt{11}}{- 10}}$

Step 8.1.2

Multiply $\sqrt{11}$ by $1$ .

$\sqrt{\frac{1 + \sqrt{11} + \sqrt{11} \cdot 1 + \sqrt{11} \sqrt{11}}{- 10}}$

Step 8.1.3

Multiply $\sqrt{11}$ by $1$ .

$\sqrt{\frac{1 + \sqrt{11} + \sqrt{11} + \sqrt{11} \sqrt{11}}{- 10}}$

Step 8.1.4

Combine using the product rule for radicals.

$\sqrt{\frac{1 + \sqrt{11} + \sqrt{11} + \sqrt{11 \cdot 11}}{- 10}}$

Step 8.1.5

Multiply $11$ by $11$ .

$\sqrt{\frac{1 + \sqrt{11} + \sqrt{11} + \sqrt{121}}{- 10}}$

Step 8.1.6

Rewrite $121$ as $11^{2}$ .

$\sqrt{\frac{1 + \sqrt{11} + \sqrt{11} + \sqrt{11^{2}}}{- 10}}$

Step 8.1.7

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

$\sqrt{\frac{1 + \sqrt{11} + \sqrt{11} + 11}{- 10}}$

Step 8.2

Add $1$ and $11$ .

$\sqrt{\frac{12 + \sqrt{11} + \sqrt{11}}{- 10}}$

Step 8.3

Add $\sqrt{11}$ and $\sqrt{11}$ .

$\sqrt{\frac{12 + 2 \sqrt{11}}{- 10}}$

Step 9

Cancel the common factor of $12 + 2 \sqrt{11}$ and $- 10$ .

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Step 9.1

Factor $2$ out of $12$ .

$\sqrt{\frac{2 \cdot 6 + 2 \sqrt{11}}{- 10}}$

Step 9.2

Factor $2$ out of $2 \sqrt{11}$ .

$\sqrt{\frac{2 \cdot 6 + 2 (\sqrt{11})}{- 10}}$

Step 9.3

Factor $2$ out of $2 \cdot 6 + 2 (\sqrt{11})$ .

$\sqrt{\frac{2 \cdot (6 + \sqrt{11})}{- 10}}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $2$ out of $- 10$ .

$\sqrt{\frac{2 \cdot (6 + \sqrt{11})}{2 (- 5)}}$

Step 9.4.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot (6 + \sqrt{11})}{2 \cdot - 5}}$

Step 9.4.3

Rewrite the expression.

$\sqrt{\frac{6 + \sqrt{11}}{- 5}}$

Step 10

Simplify the expression.

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Step 10.1

Move the negative in front of the fraction.

$\sqrt{- \frac{6 + \sqrt{11}}{5}}$

Step 10.2

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{6 + \sqrt{11}}{5}}$

Step 11

Pull terms out from under the radical.

$i \sqrt{\frac{6 + \sqrt{11}}{5}}$

Step 12

Rewrite $\sqrt{\frac{6 + \sqrt{11}}{5}}$ as $\frac{\sqrt{6 + \sqrt{11}}}{\sqrt{5}}$ .

$i \frac{\sqrt{6 + \sqrt{11}}}{\sqrt{5}}$

Step 13

Multiply $\frac{\sqrt{6 + \sqrt{11}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$i (\frac{\sqrt{6 + \sqrt{11}}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 14

Simplify terms.

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Step 14.1

Combine and simplify the denominator.

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Step 14.1.1

Multiply $\frac{\sqrt{6 + \sqrt{11}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 14.1.2

Raise $\sqrt{5}$ to the power of $1$ .

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 14.1.3

Raise $\sqrt{5}$ to the power of $1$ .

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 14.1.4

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

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 14.1.5

Add $1$ and $1$ .

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 14.1.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 14.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 14.1.6.2

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

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 14.1.6.3

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

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 14.1.6.4

Cancel the common factor of $2$ .

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Step 14.1.6.4.1

Cancel the common factor.

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 14.1.6.4.2

Rewrite the expression.

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{5^{1}}$

Step 14.1.6.5

Evaluate the exponent.

$i \frac{\sqrt{6 + \sqrt{11}} \sqrt{5}}{5}$

Step 14.2

Combine using the product rule for radicals.

$i \frac{\sqrt{(6 + \sqrt{11}) \cdot 5}}{5}$

Step 14.3

Combine $i$ and $\frac{\sqrt{(6 + \sqrt{11}) \cdot 5}}{5}$ .

$\frac{i \sqrt{(6 + \sqrt{11}) \cdot 5}}{5}$

Step 15

Simplify the numerator.

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Step 15.1

Apply the distributive property.

$\frac{i \sqrt{6 \cdot 5 + \sqrt{11} \cdot 5}}{5}$

Step 15.2

Multiply $6$ by $5$ .

$\frac{i \sqrt{30 + \sqrt{11} \cdot 5}}{5}$

Step 15.3

Move $5$ to the left of $\sqrt{11}$ .

$\frac{i \sqrt{30 + 5 \sqrt{11}}}{5}$