Trigonometry Examples

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

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.

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

Step 1.2

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

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

Step 1.3

Combine exponents.

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Step 1.3.1

Factor out negative.

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

Step 1.3.2

Combine $2$ and $\frac{\sqrt{3}}{3}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

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 $b = \tan (\frac{5 π}{6})$ .

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

Step 2.3

Simplify.

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Step 2.3.1

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.

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

Step 2.3.2

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

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

Step 2.3.3

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.

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

Step 2.3.4

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

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

Step 2.3.5

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

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Step 2.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.5.2

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

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

Step 2.4

Write $1$ as a fraction with a common denominator.

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Write $1$ as a fraction with a common denominator.

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $3$ by $3$ .

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

Step 4.2.1.2

Multiply $3$ by $- 1$ .

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

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

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

Step 4.2.1.3.2

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

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

Step 4.2.1.3.3

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

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

Step 4.2.1.3.4

Add $1$ and $1$ .

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

Step 4.2.1.4

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

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Step 4.2.1.4.1

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

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

Step 4.2.1.4.2

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

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

Step 4.2.1.4.3

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

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

Step 4.2.1.4.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.4.1

Cancel the common factor.

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

Step 4.2.1.4.4.2

Rewrite the expression.

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

Step 4.2.1.4.5

Evaluate the exponent.

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

Step 4.2.1.5

Multiply $- 1$ by $3$ .

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

Step 4.2.2

Subtract $3$ from $9$ .

$\frac{- \frac{2 \sqrt{3}}{3}}{\frac{6 + 3 \sqrt{3} - 3 \sqrt{3}}{3 \cdot 3}}$

Step 4.2.3

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

$\frac{- \frac{2 \sqrt{3}}{3}}{\frac{6 + 0}{3 \cdot 3}}$

Step 4.2.4

Add $6$ and $0$ .

$\frac{- \frac{2 \sqrt{3}}{3}}{\frac{6}{3 \cdot 3}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Multiply $3$ by $3$ .

$\frac{- \frac{2 \sqrt{3}}{3}}{\frac{6}{9}}$

Step 5.2

Cancel the common factor of $6$ and $9$ .

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Step 5.2.1

Factor $3$ out of $6$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $3$ out of $9$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Cancel the common factor.

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

Step 7.4

Rewrite the expression.

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

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$- \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$