Trigonometry Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Expand the denominator using the FOIL method.

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

Step 4.3

Simplify.

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

Step 4.4

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

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

Step 5

Expand $(2 - \sqrt{3}) (2 - \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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 $2$ by $2$ .

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

Step 6.1.2

Multiply $- 1$ by $2$ .

$v (4 - 2 \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Step 6.1.3

Multiply $2$ by $- 1$ .

$v (4 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{3}))$

Step 6.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.4.2

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

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

Step 6.1.4.3

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

$v (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3})$

Step 6.1.4.4

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

$v (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 6.1.4.5

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

$v (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1 + 1})$

Step 6.1.4.6

Add $1$ and $1$ .

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

Step 6.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 6.1.5.2

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

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

Step 6.1.5.3

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

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

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

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

Step 6.1.5.4.2

Rewrite the expression.

$v (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{1})$

Step 6.1.5.5

Evaluate the exponent.

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

Step 6.2

Add $4$ and $3$ .

$v (7 - 2 \sqrt{3} - 2 \sqrt{3})$

Step 6.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$v (7 - 4 \sqrt{3})$

Step 7

Apply the distributive property.

$v \cdot 7 + v (- 4 \sqrt{3})$

Step 8

Simplify the expression.

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Step 8.1

Move $7$ to the left of $v$ .

$7 \cdot v + v (- 4 \sqrt{3})$

Step 8.2

Reorder factors in $7 v + v (- 4 \sqrt{3})$ .

$7 v - 4 v \sqrt{3}$