Trigonometry Examples

$\tan (x + 30)$

Step 1

Double negation

$\tan (x - (- 30))$

Step 2

Apply the difference of angles identity.

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

Step 3

Simplify the numerator.

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Step 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 fourth quadrant.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2

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

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

Step 4

Simplify the denominator.

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Step 4.1

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 fourth quadrant.

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Combine.

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

Step 6

Apply the distributive property.

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

Step 7

Simplify by cancelling.

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Step 7.1

Cancel the common factor of $3$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

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

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Multiply $3$ by $1$ .

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

Step 9

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

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

Step 10

Simplify terms.

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Step 10.1

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

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

Step 10.2

Expand the denominator using the FOIL method.

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

Step 10.3

Simplify.

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Step 10.3.1

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

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Step 10.3.1.1

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

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

Step 10.3.1.2

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

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

Step 10.3.1.3

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

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

Step 10.3.1.4

Cancel the common factor of $2$ .

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Step 10.3.1.4.1

Cancel the common factor.

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

Step 10.3.1.4.2

Rewrite the expression.

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

Step 10.3.1.5

Evaluate the exponent.

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

Step 10.3.2

Multiply $3$ by $- 1$ .

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

Step 10.4

Factor $3$ out of $9 - 3 \tan^{2} (x)$ .

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Step 10.4.1

Factor $3$ out of $9$ .

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

Step 10.4.2

Factor $3$ out of $- 3 \tan^{2} (x)$ .

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

Step 10.4.3

Factor $3$ out of $3 (3) + 3 (- \tan^{2} (x))$ .

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

Step 11

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

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Apply the distributive property.

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

Step 12

Simplify and combine like terms.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply $3$ by $3$ .

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

Step 12.1.2

Multiply $3 \tan (x) (\tan (x) \sqrt{3})$ .

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Step 12.1.2.1

Raise $\tan (x)$ to the power of $1$ .

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

Step 12.1.2.2

Raise $\tan (x)$ to the power of $1$ .

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

Step 12.1.2.3

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

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

Step 12.1.2.4

Add $1$ and $1$ .

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

Step 12.1.3

Move $3$ to the left of $\sqrt{3}$ .

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

Step 12.1.4

Multiply $\sqrt{3} (\tan (x) \sqrt{3})$ .

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Step 12.1.4.1

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

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

Step 12.1.4.2

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

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

Step 12.1.4.3

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

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

Step 12.1.4.4

Add $1$ and $1$ .

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

Step 12.1.5

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

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Step 12.1.5.1

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

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

Step 12.1.5.2

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

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

Step 12.1.5.3

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

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

Step 12.1.5.4

Cancel the common factor of $2$ .

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Step 12.1.5.4.1

Cancel the common factor.

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

Step 12.1.5.4.2

Rewrite the expression.

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

Step 12.1.5.5

Evaluate the exponent.

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

Step 12.2

Add $9 \tan (x)$ and $3 \tan (x)$ .

$\frac{12 \tan (x) + 3 \tan^{2} (x) \sqrt{3} + 3 \sqrt{3}}{3 (3 - \tan^{2} (x))}$

Step 13

Cancel the common factor of $12 \tan (x) + 3 \tan^{2} (x) \sqrt{3} + 3 \sqrt{3}$ and $3$ .

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Step 13.1

Factor $3$ out of $12 \tan (x)$ .

$\frac{3 (4 \tan (x)) + 3 \tan^{2} (x) \sqrt{3} + 3 \sqrt{3}}{3 (3 - \tan^{2} (x))}$

Step 13.2

Factor $3$ out of $3 \tan^{2} (x) \sqrt{3}$ .

$\frac{3 (4 \tan (x)) + 3 (\tan^{2} (x) \sqrt{3}) + 3 \sqrt{3}}{3 (3 - \tan^{2} (x))}$

Step 13.3

Factor $3$ out of $3 (4 \tan (x)) + 3 (\tan^{2} (x) \sqrt{3})$ .

$\frac{3 (4 \tan (x) + \tan^{2} (x) \sqrt{3}) + 3 \sqrt{3}}{3 (3 - \tan^{2} (x))}$

Step 13.4

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

$\frac{3 (4 \tan (x) + \tan^{2} (x) \sqrt{3}) + 3 (\sqrt{3})}{3 (3 - \tan^{2} (x))}$

Step 13.5

Factor $3$ out of $3 (4 \tan (x) + \tan^{2} (x) \sqrt{3}) + 3 (\sqrt{3})$ .

$\frac{3 (4 \tan (x) + \tan^{2} (x) \sqrt{3} + \sqrt{3})}{3 (3 - \tan^{2} (x))}$

Step 13.6

Cancel the common factors.

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Step 13.6.1

Cancel the common factor.

$\frac{3 (4 \tan (x) + \tan^{2} (x) \sqrt{3} + \sqrt{3})}{3 (3 - \tan^{2} (x))}$

Step 13.6.2

Rewrite the expression.

$\frac{4 \tan (x) + \tan^{2} (x) \sqrt{3} + \sqrt{3}}{3 - \tan^{2} (x)}$