Precalculus Examples

$\frac{2 \tan (150)}{1 - \tan^{2} (150)}$

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.

$\frac{2 (- \tan (30))}{1 - \tan^{2} (150)}$

Step 1.2

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

$\frac{2 \cdot - 1 \frac{\sqrt{3}}{3}}{1 - \tan^{2} (150)}$

Step 1.3

Combine exponents.

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Step 1.3.1

Factor out negative.

$\frac{- (2 \frac{\sqrt{3}}{3})}{1 - \tan^{2} (150)}$

Step 1.3.2

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

$\frac{- \frac{2 \sqrt{3}}{3}}{1 - \tan^{2} (150)}$

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.

$\frac{- \frac{2 \sqrt{3}}{3}}{1 - {(- \tan (30))}^{2}}$

Step 2.2

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

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

Step 2.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.1

Apply the product rule to $- \frac{\sqrt{3}}{3}$ .

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

Step 2.3.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

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

Step 2.4

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.4.1

Move ${(- 1)}^{2}$ .

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

Step 2.4.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.4.2.1

Raise $- 1$ to the power of $1$ .

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

Step 2.4.2.2

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

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

Step 2.4.3

Add $2$ and $1$ .

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

Step 2.5

Raise $- 1$ to the power of $3$ .

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

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 $3^{\frac{1}{2}}$ .

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 2.7

Raise $3$ to the power of $2$ .

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

Step 2.8

Cancel the common factor of $3$ and $9$ .

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Step 2.8.1

Factor $3$ out of $3$ .

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

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $3$ out of $9$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.9

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

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

Step 2.10

Combine the numerators over the common denominator.

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

Step 2.11

Subtract $1$ from $3$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

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.

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

Step 4.2

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

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

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

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$- \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$