Precalculus Examples

$\frac{\tan (80) - \tan (30)}{1 + \tan (80) \tan (30)}$

Step 1

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\tan (80) - \frac{\sqrt{3}}{3}}{1 + \tan (80) \tan (30)}$

Step 2

Simplify the denominator.

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Step 2.1

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\tan (80) - \frac{\sqrt{3}}{3}}{1 + \tan (80) \frac{\sqrt{3}}{3}}$

Step 2.2

Combine $\tan (80)$ and $\frac{\sqrt{3}}{3}$ .

$\frac{\tan (80) - \frac{\sqrt{3}}{3}}{1 + \frac{\tan (80) \sqrt{3}}{3}}$

Step 3

Multiply the numerator and denominator of the fraction by $3$ .

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Step 3.1

Multiply $\frac{\tan (80) - \frac{\sqrt{3}}{3}}{1 + \frac{\tan (80) \sqrt{3}}{3}}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{\tan (80) - \frac{\sqrt{3}}{3}}{1 + \frac{\tan (80) \sqrt{3}}{3}}$

Step 3.2

Combine.

$\frac{3 (\tan (80) - \frac{\sqrt{3}}{3})}{3 (1 + \frac{\tan (80) \sqrt{3}}{3})}$

Step 4

Apply the distributive property.

$\frac{3 \tan (80) + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 \frac{\tan (80) \sqrt{3}}{3}}$

Step 5

Simplify by cancelling.

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Step 5.1

Cancel the common factor of $3$ .

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Step 5.1.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{3 \tan (80) + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 \frac{\tan (80) \sqrt{3}}{3}}$

Step 5.1.2

Cancel the common factor.

$\frac{3 \tan (80) + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 \frac{\tan (80) \sqrt{3}}{3}}$

Step 5.1.3

Rewrite the expression.

$\frac{3 \tan (80) - \sqrt{3}}{3 \cdot 1 + 3 \frac{\tan (80) \sqrt{3}}{3}}$

Step 5.2

Cancel the common factor of $3$ .

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Step 5.2.1

Cancel the common factor.

$\frac{3 \tan (80) - \sqrt{3}}{3 \cdot 1 + 3 \frac{\tan (80) \sqrt{3}}{3}}$

Step 5.2.2

Rewrite the expression.

$\frac{3 \tan (80) - \sqrt{3}}{3 \cdot 1 + \tan (80) \sqrt{3}}$

Step 6

Multiply $3$ by $1$ .

$\frac{3 \tan (80) - \sqrt{3}}{3 + \tan (80) \sqrt{3}}$

Step 7

Multiply $\frac{3 \tan (80) - \sqrt{3}}{3 + \tan (80) \sqrt{3}}$ by $\frac{3 - \tan (80) \sqrt{3}}{3 - \tan (80) \sqrt{3}}$ .

$\frac{3 \tan (80) - \sqrt{3}}{3 + \tan (80) \sqrt{3}} \cdot \frac{3 - \tan (80) \sqrt{3}}{3 - \tan (80) \sqrt{3}}$

Step 8

Combine fractions.

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Step 8.1

Multiply $\frac{3 \tan (80) - \sqrt{3}}{3 + \tan (80) \sqrt{3}}$ by $\frac{3 - \tan (80) \sqrt{3}}{3 - \tan (80) \sqrt{3}}$ .

$\frac{(3 \tan (80) - \sqrt{3}) (3 - \tan (80) \sqrt{3})}{(3 + \tan (80) \sqrt{3}) (3 - \tan (80) \sqrt{3})}$

Step 8.2

Expand the denominator using the FOIL method.

$\frac{(3 \tan (80) - \sqrt{3}) (3 - \tan (80) \sqrt{3})}{9 - 3 (\tan (80) \sqrt{3}) + \tan (80) \sqrt{3} \cdot 3 - \tan^{2} (80) {\sqrt{3}}^{2}}$

Step 8.3

Simplify.

$\frac{(3 \tan (80) - \sqrt{3}) (3 - \tan (80) \sqrt{3})}{9 - 3 \tan^{2} (80)}$

Step 9

Simplify each term.

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Step 9.1

Evaluate $\tan (80)$ .

$\frac{(3 \cdot 5.67128181 - \sqrt{3}) (3 - \tan (80) \sqrt{3})}{9 - 3 \tan^{2} (80)}$

Step 9.2

Multiply $3$ by $5.67128181$ .

$\frac{(17.01384545 - \sqrt{3}) (3 - \tan (80) \sqrt{3})}{9 - 3 \tan^{2} (80)}$

Step 10

Subtract $\sqrt{3}$ from $17.01384545$ .

$\frac{15.28179465 (3 - \tan (80) \sqrt{3})}{9 - 3 \tan^{2} (80)}$

Step 11

Simplify each term.

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Step 11.1

Evaluate $\tan (80)$ .

$\frac{15.28179465 (3 - 1 \cdot 5.67128181 \sqrt{3})}{9 - 3 \tan^{2} (80)}$

Step 11.2

Multiply $- 1$ by $5.67128181$ .

$\frac{15.28179465 (3 - 5.67128181 \sqrt{3})}{9 - 3 \tan^{2} (80)}$

Step 11.3

Multiply $- 5.67128181$ by $\sqrt{3}$ .

$\frac{15.28179465 (3 - 9.82294825)}{9 - 3 \tan^{2} (80)}$

Step 12

Simplify the expression.

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Step 12.1

Subtract $9.82294825$ from $3$ .

$\frac{15.28179465 \cdot - 6.82294825}{9 - 3 \tan^{2} (80)}$

Step 12.2

Multiply $15.28179465$ by $- 6.82294825$ .

$\frac{- 104.26689415}{9 - 3 \tan^{2} (80)}$

Step 12.3

Move the negative in front of the fraction.

$- \frac{104.26689415}{9 - 3 \tan^{2} (80)}$

Step 13

Evaluate $\tan (80)$ .

$- \frac{104.26689415}{9 - 3 \cdot {5.67128181}^{2}}$

Step 14

Raise $5.67128181$ to the power of $2$ .

$- \frac{104.26689415}{9 - 3 \cdot 32.16343747}$

Step 15

Multiply $- 3$ by $32.16343747$ .

$- \frac{104.26689415}{9 - 96.49031243}$

Step 16

Subtract $96.49031243$ from $9$ .

$- \frac{104.26689415}{- 87.49031243}$

Step 17

Divide $104.26689415$ by $- 87.49031243$ .

$- - 1.19175359$

Step 18

Multiply $- 1$ by $- 1.19175359$ .

$1.19175359$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{\tan (80) - \tan (30)}{1 + \tan (80) \tan (30)}$

Decimal Form:

$1.19175359 \dots$