Algebra Examples

$\frac{\tan (\frac{2 π}{3}) - \tan (\frac{5 π}{6})}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- \tan (\frac{π}{3}) - \tan (\frac{5 π}{6})}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.2

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

$\frac{- \sqrt{3} - \tan (\frac{5 π}{6})}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- \sqrt{3} - - \tan (\frac{π}{6})}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.4

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{- \sqrt{3} - - \frac{\sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.5

Multiply $- - \frac{\sqrt{3}}{3}$ .

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Step 1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{- \sqrt{3} + 1 \frac{\sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.5.2

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

$\frac{- \sqrt{3} + \frac{\sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.6

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- \sqrt{3} \cdot \frac{3}{3} + \frac{\sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.7

Combine $- \sqrt{3}$ and $\frac{3}{3}$ .

$\frac{\frac{- \sqrt{3} \cdot 3}{3} + \frac{\sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.8

Combine the numerators over the common denominator.

$\frac{\frac{- \sqrt{3} \cdot 3 + \sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.9

Rewrite $\frac{- \sqrt{3} \cdot 3 + \sqrt{3}}{3}$ in a factored form.

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Step 1.9.1

Multiply $3$ by $- 1$ .

$\frac{\frac{- 3 \sqrt{3} + \sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.9.2

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

$\frac{\frac{- 2 \sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 1.10

Move the negative in front of the fraction.

$\frac{- \frac{2 \sqrt{3}}{3}}{1 + \tan (\frac{2 π}{3}) \tan (\frac{5 π}{6})}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- \frac{2 \sqrt{3}}{3}}{1 - \tan (\frac{π}{3}) \tan (\frac{5 π}{6})}$

Step 2.2

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

$\frac{- \frac{2 \sqrt{3}}{3}}{1 - \sqrt{3} \tan (\frac{5 π}{6})}$

Step 2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- \frac{2 \sqrt{3}}{3}}{1 - \sqrt{3} (- \tan (\frac{π}{6}))}$

Step 2.4

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

Step 2.5

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

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Step 2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.2

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

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

Step 2.5.3

Combine $\sqrt{3}$ and $\frac{\sqrt{3}}{3}$ .

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

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

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

Step 2.5.7

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 2.7

Divide $3$ by $3$ .

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

Step 2.8

Add $1$ and $1$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Move the leading negative in $- \frac{2 \sqrt{3}}{3}$ into the numerator.

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

Step 4.2

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

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

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

Step 5

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.57735026 \dots$