Trigonometry Examples

\tan (\frac{13 π}{12})

$\tan (\frac{13 π}{12})$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

\tan (\frac{π}{12})

$\tan (\frac{π}{12})$

Step 2

Split

\frac{π}{12}

$\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

\tan (\frac{π}{4} - \frac{π}{6})

$\tan (\frac{π}{4} - \frac{π}{6})$

Step 3

Apply the difference of angles identity.

\frac{\tan (\frac{π}{4}) - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}

$\frac{\tan (\frac{π}{4}) - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 4

The exact value of

\tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

\frac{1 - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}

$\frac{1 - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 5

The exact value of

\tan (\frac{π}{6})

$\tan (\frac{π}{6})$ is

\frac{\sqrt{3}}{3}

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

\frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}

$\frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 6

The exact value of

\tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

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

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

Step 7

The exact value of

\tan (\frac{π}{6})

$\tan (\frac{π}{6})$ is

\frac{\sqrt{3}}{3}

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

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

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

Step 8

Simplify

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

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

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Step 8.1

Multiply the numerator and denominator of the fraction by

3

$3$ .

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Step 8.1.1

Multiply

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

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

\frac{3}{3}

$\frac{3}{3}$ .

\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}

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

Step 8.1.2

Combine.

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

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

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

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

Step 8.2

Apply the distributive property.

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

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

Step 8.3

Cancel the common factor of

3

$3$ .

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Step 8.3.1

Move the leading negative in

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

$- \frac{\sqrt{3}}{3}$ into the numerator.

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

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

Step 8.3.2

Cancel the common factor.

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

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

Step 8.3.3

Rewrite the expression.

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

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

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

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

Step 8.4

Multiply

3

$3$ by

1

$1$ .

\frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}}

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

Step 8.5

Simplify the denominator.

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Step 8.5.1

Multiply

3

$3$ by

1

$1$ .

\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}

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

Step 8.5.2

Cancel the common factor of

3

$3$ .

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Step 8.5.2.1

Factor

3

$3$ out of

3 \cdot 1

$3 \cdot 1$ .

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

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

Step 8.5.2.2

Cancel the common factor.

\frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}

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

Step 8.5.2.3

Rewrite the expression.

\frac{3 - \sqrt{3}}{3 + \sqrt{3}}

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

\frac{3 - \sqrt{3}}{3 + \sqrt{3}}

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

\frac{3 - \sqrt{3}}{3 + \sqrt{3}}

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

Step 8.6

Multiply

\frac{3 - \sqrt{3}}{3 + \sqrt{3}}

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

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

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

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

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

Step 8.7

Multiply

\frac{3 - \sqrt{3}}{3 + \sqrt{3}}

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

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

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

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

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

Step 8.8

Expand the denominator using the FOIL method.

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

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

Step 8.9

Simplify.

\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 8.10

Simplify the numerator.

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Step 8.10.1

Raise

3 - \sqrt{3}

$3 - \sqrt{3}$ to the power of

1

$1$ .

\frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}

$\frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}$

Step 8.10.2

Raise

3 - \sqrt{3}

$3 - \sqrt{3}$ to the power of

1

$1$ .

\frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}

$\frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}$

Step 8.10.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{{(3 - \sqrt{3})}^{1 + 1}}{6}

$\frac{{(3 - \sqrt{3})}^{1 + 1}}{6}$

Step 8.10.4

Add

1

$1$ and

1

$1$ .

\frac{{(3 - \sqrt{3})}^{2}}{6}

$\frac{{(3 - \sqrt{3})}^{2}}{6}$

\frac{{(3 - \sqrt{3})}^{2}}{6}

$\frac{{(3 - \sqrt{3})}^{2}}{6}$

Step 8.11

Rewrite

{(3 - \sqrt{3})}^{2}

${(3 - \sqrt{3})}^{2}$ as

(3 - \sqrt{3}) (3 - \sqrt{3})

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

\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 8.12

Expand

(3 - \sqrt{3}) (3 - \sqrt{3})

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

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Step 8.12.1

Apply the distributive property.

\frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}

$\frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 8.12.2

Apply the distributive property.

\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}

$\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 8.12.3

Apply the distributive property.

\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}

$\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}

$\frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 8.13

Simplify and combine like terms.

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Step 8.13.1

Simplify each term.

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Step 8.13.1.1

Multiply

3

$3$ by

3

$3$ .

\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}

$\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 8.13.1.2

Multiply

- 1

$- 1$ by

3

$3$ .

\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}

$\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 8.13.1.3

Multiply

3

$3$ by

- 1

$- 1$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

Step 8.13.1.4

Multiply

- \sqrt{3} (- \sqrt{3})

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

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Step 8.13.1.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

Step 8.13.1.4.2

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}

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

Step 8.13.1.4.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

Step 8.13.1.4.4

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

Step 8.13.1.4.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Step 8.13.1.4.6

Add

1

$1$ and

1

$1$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}

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

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}

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

Step 8.13.1.5

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 8.13.1.5.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

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

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

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}$

Step 8.13.1.5.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}$

Step 8.13.1.5.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}

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

Step 8.13.1.5.4

Cancel the common factor of

2

$2$ .

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Step 8.13.1.5.4.1

Cancel the common factor.

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}

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

Step 8.13.1.5.4.2

Rewrite the expression.

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Step 8.13.1.5.5

Evaluate the exponent.

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}

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

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}

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

\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}

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

Step 8.13.2

Add

9

$9$ and

3

$3$ .

\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}

$\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}$

Step 8.13.3

Subtract

3 \sqrt{3}

$3 \sqrt{3}$ from

- 3 \sqrt{3}

$- 3 \sqrt{3}$ .

\frac{12 - 6 \sqrt{3}}{6}

$\frac{12 - 6 \sqrt{3}}{6}$

\frac{12 - 6 \sqrt{3}}{6}

$\frac{12 - 6 \sqrt{3}}{6}$

Step 8.14

Cancel the common factor of

12 - 6 \sqrt{3}

$12 - 6 \sqrt{3}$ and

6

$6$ .

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Step 8.14.1

Factor

6

$6$ out of

12

$12$ .

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

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

Step 8.14.2

Factor

6

$6$ out of

- 6 \sqrt{3}

$- 6 \sqrt{3}$ .

\frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}

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

Step 8.14.3

Factor

6

$6$ out of

6 (2) + 6 (- \sqrt{3})

$6 (2) + 6 (- \sqrt{3})$ .

\frac{6 (2 - \sqrt{3})}{6}

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

Step 8.14.4

Cancel the common factors.

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Step 8.14.4.1

Factor

6

$6$ out of

6

$6$ .

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

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

Step 8.14.4.2

Cancel the common factor.

\frac{6 (2 - \sqrt{3})}{6 \cdot 1}

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

Step 8.14.4.3

Rewrite the expression.

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

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

Step 8.14.4.4

Divide

2 - \sqrt{3}

$2 - \sqrt{3}$ by

1

$1$ .

2 - \sqrt{3}

$2 - \sqrt{3}$

2 - \sqrt{3}

$2 - \sqrt{3}$

2 - \sqrt{3}

$2 - \sqrt{3}$

2 - \sqrt{3}

$2 - \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

2 - \sqrt{3}

$2 - \sqrt{3}$

Decimal Form:

0.26794919 \dots

$0.26794919 \dots$