Trigonometry Examples

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

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify ${(\sqrt{3} - 1)}^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Combine using the product rule for radicals.

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

Step 2.3.1.2

Multiply $3$ by $3$ .

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

Step 2.3.1.3

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

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Move $- 1$ to the left of $\sqrt{3}$ .

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

Step 2.3.1.6

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 2.3.1.7

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 2.3.1.8

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2

Add $3$ and $1$ .

$2 \frac{1}{4 - \sqrt{3} - \sqrt{3}}$

Step 2.3.3

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$2 \frac{1}{4 - 2 \sqrt{3}}$

Step 3

Multiply $\frac{1}{4 - 2 \sqrt{3}}$ by $\frac{4 + 2 \sqrt{3}}{4 + 2 \sqrt{3}}$ .

$2 (\frac{1}{4 - 2 \sqrt{3}} \cdot \frac{4 + 2 \sqrt{3}}{4 + 2 \sqrt{3}})$

Step 4

Multiply $\frac{1}{4 - 2 \sqrt{3}}$ by $\frac{4 + 2 \sqrt{3}}{4 + 2 \sqrt{3}}$ .

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

Step 5

Expand the denominator using the FOIL method.

$2 \frac{4 + 2 \sqrt{3}}{16 + 8 \sqrt{3} - 8 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 6

Simplify.

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $4$ .

$2 \frac{4 + 2 \sqrt{3}}{2 (2)}$

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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Step 8.1

Factor $2$ out of $4$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $2$ out of $2$ .

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

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

Step 8.4.4

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

$2 + \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$