Trigonometry Examples

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

Step 1

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \tan (\frac{π}{3})$ .

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

Step 2.3

Simplify.

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Step 2.3.1

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

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

Step 2.3.2

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $1$ by $1$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

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

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

Step 3.2.1.4.3

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

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

Step 3.2.1.4.4

Add $1$ and $1$ .

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

Step 3.2.1.5

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

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Step 3.2.1.5.1

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

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

Step 3.2.1.5.2

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

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

Step 3.2.1.5.3

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

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

Step 3.2.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.5.4.1

Cancel the common factor.

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

Step 3.2.1.5.4.2

Rewrite the expression.

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

Step 3.2.1.5.5

Evaluate the exponent.

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

Step 3.2.1.6

Multiply $- 1$ by $3$ .

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

Step 3.2.2

Subtract $3$ from $1$ .

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

Step 3.2.3

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

$\frac{2 \sqrt{3}}{- 2 + 0}$

Step 3.2.4

Add $- 2$ and $0$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $2$ and $- 2$ .

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Step 4.1.1

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

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

Step 4.1.2

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

$- 1 \cdot \sqrt{3}$

Step 4.2

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

$- \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$