Trigonometry Examples

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

Step 1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \sqrt{\frac{1 - - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}{1 - \frac{1}{\sqrt{2}}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Simplify the expression.

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Step 8.1

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

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

Step 8.2

Combine the numerators over the common denominator.

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

Step 9

Multiply the numerator by the reciprocal of the denominator.

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

Multiply $\frac{1}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

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

Step 12

Multiply $\frac{1}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$- \sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{(2 - \sqrt{2}) (2 + \sqrt{2})}}$

Step 13

Expand the denominator using the FOIL method.

$- \sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}}$

Step 14

Simplify.

$- \sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{2}}$

Step 15

Apply the distributive property.

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

Step 16

Cancel the common factor of $2$ .

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Step 16.1

Cancel the common factor.

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

Step 16.2

Rewrite the expression.

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

Step 17

Combine $\sqrt{2}$ and $\frac{2 + \sqrt{2}}{2}$ .

$- \sqrt{2 + \sqrt{2} + \frac{\sqrt{2} (2 + \sqrt{2})}{2}}$

Step 18

Find the common denominator.

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Step 18.1

Write $2$ as a fraction with denominator $1$ .

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

Step 18.2

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

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

Step 18.3

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

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

Step 18.4

Write $\sqrt{2}$ as a fraction with denominator $1$ .

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

Step 18.5

Multiply $\frac{\sqrt{2}}{1}$ by $\frac{2}{2}$ .

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

Step 18.6

Multiply $\frac{\sqrt{2}}{1}$ by $\frac{2}{2}$ .

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

Step 19

Combine the numerators over the common denominator.

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

Step 20

Simplify each term.

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Step 20.1

Multiply $2$ by $2$ .

$- \sqrt{\frac{4 + \sqrt{2} \cdot 2 + \sqrt{2} (2 + \sqrt{2})}{2}}$

Step 20.2

Move $2$ to the left of $\sqrt{2}$ .

$- \sqrt{\frac{4 + 2 \cdot \sqrt{2} + \sqrt{2} (2 + \sqrt{2})}{2}}$

Step 20.3

Apply the distributive property.

$- \sqrt{\frac{4 + 2 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{2}}$

Step 20.4

Move $2$ to the left of $\sqrt{2}$ .

$- \sqrt{\frac{4 + 2 \sqrt{2} + 2 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{2}}$

Step 20.5

Combine using the product rule for radicals.

$- \sqrt{\frac{4 + 2 \sqrt{2} + 2 \cdot \sqrt{2} + \sqrt{2 \cdot 2}}{2}}$

Step 20.6

Simplify each term.

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Step 20.6.1

Multiply $2$ by $2$ .

$- \sqrt{\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{4}}{2}}$

Step 20.6.2

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

$- \sqrt{\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{2^{2}}}{2}}$

Step 20.6.3

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

$- \sqrt{\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + 2}{2}}$

Step 21

Simplify terms.

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Step 21.1

Add $4$ and $2$ .

$- \sqrt{\frac{6 + 2 \sqrt{2} + 2 \sqrt{2}}{2}}$

Step 21.2

Add $2 \sqrt{2}$ and $2 \sqrt{2}$ .

$- \sqrt{\frac{6 + 4 \sqrt{2}}{2}}$

Step 21.3

Cancel the common factor of $6 + 4 \sqrt{2}$ and $2$ .

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Step 21.3.1

Factor $2$ out of $6$ .

$- \sqrt{\frac{2 \cdot 3 + 4 \sqrt{2}}{2}}$

Step 21.3.2

Factor $2$ out of $4 \sqrt{2}$ .

$- \sqrt{\frac{2 \cdot 3 + 2 (2 \sqrt{2})}{2}}$

Step 21.3.3

Factor $2$ out of $2 (3) + 2 (2 \sqrt{2})$ .

$- \sqrt{\frac{2 (3 + 2 \sqrt{2})}{2}}$

Step 21.3.4

Cancel the common factors.

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Step 21.3.4.1

Factor $2$ out of $2$ .

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

Step 21.3.4.2

Cancel the common factor.

$- \sqrt{\frac{2 (3 + 2 \sqrt{2})}{2 \cdot 1}}$

Step 21.3.4.3

Rewrite the expression.

$- \sqrt{\frac{3 + 2 \sqrt{2}}{1}}$

Step 21.3.4.4

Divide $3 + 2 \sqrt{2}$ by $1$ .

$- \sqrt{3 + 2 \sqrt{2}}$

Step 22

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3 + 2 \sqrt{2}}$

Decimal Form:

$- 2.41421356 \dots$