Trigonometry Examples

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

Step 1

Multiply $\frac{1 + \sqrt{3}}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 + \sqrt{3}}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Raise $1 + \sqrt{3}$ to the power of $1$ .

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

Step 3.2

Raise $1 + \sqrt{3}$ to the power of $1$ .

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

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

Step 5

Expand $(1 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $1$ by $1$ .

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

Step 6.1.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 6.1.3

Multiply $\sqrt{3}$ by $1$ .

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

Step 6.1.4

Combine using the product rule for radicals.

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

Step 6.1.5

Multiply $3$ by $3$ .

$\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{9}}{- 2}$

Step 6.1.6

Rewrite $9$ as $3^{2}$ .

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

Step 6.1.7

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

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

Step 6.2

Add $1$ and $3$ .

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

Step 6.3

Add $\sqrt{3}$ and $\sqrt{3}$ .

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

Step 7

Cancel the common factor of $4 + 2 \sqrt{3}$ and $- 2$ .

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Step 7.1

Factor $2$ out of $4$ .

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

Step 7.2

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 7.3

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

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

Step 7.4

Move the negative one from the denominator of $\frac{2 + \sqrt{3}}{- 1}$ .

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

Step 8

Rewrite $- 1 \cdot (2 + \sqrt{3})$ as $- (2 + \sqrt{3})$ .

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

Step 9

Apply the distributive property.

$- 1 \cdot 2 - \sqrt{3}$

Step 10

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$