Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Simplify terms.

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Step 6.1

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

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

Step 6.2

Expand the denominator using the FOIL method.

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

Step 6.3

Simplify.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Cancel the common factor of $2$ .

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Step 6.5.1

Cancel the common factor.

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

Step 6.5.2

Rewrite the expression.

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

Step 6.6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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

Step 7.3

Combine using the product rule for radicals.

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

Step 7.4

Simplify each term.

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Step 7.4.1

Multiply $2$ by $2$ .

$v (2 + \sqrt{2} + \frac{2 \sqrt{2} + \sqrt{4}}{2})$

Step 7.4.2

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

$v (2 + \sqrt{2} + \frac{2 \sqrt{2} + \sqrt{2^{2}}}{2})$

Step 7.4.3

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

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

Step 7.5

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

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Step 7.5.1

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

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

Step 7.5.2

Factor $2$ out of $2$ .

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

Step 7.5.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (1)$ .

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

Step 7.5.4

Cancel the common factors.

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Step 7.5.4.1

Factor $2$ out of $2$ .

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

Step 7.5.4.2

Cancel the common factor.

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

Step 7.5.4.3

Rewrite the expression.

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

Step 7.5.4.4

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

$v (2 + \sqrt{2} + \sqrt{2} + 1)$

Step 8

Simplify terms.

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Step 8.1

Add $2$ and $1$ .

$v (3 + \sqrt{2} + \sqrt{2})$

Step 8.2

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

$v (3 + 2 \sqrt{2})$

Step 8.3

Apply the distributive property.

$v \cdot 3 + v (2 \sqrt{2})$

Step 8.4

Move $3$ to the left of $v$ .

$3 \cdot v + v (2 \sqrt{2})$

Step 9

Move $2$ to the left of $v$ .

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