Trigonometry Examples

tan (\frac{π}{12})

$\tan (\frac{π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

\frac{π}{12}

$\frac{π}{12}$ can be split into

\frac{π}{3} - \frac{π}{4}

$\frac{π}{3} - \frac{π}{4}$ .

tan (\frac{π}{3} - \frac{π}{4})

$\tan (\frac{π}{3} - \frac{π}{4})$

Step 2

Use the difference formula for tangent to simplify the expression. The formula states that

tan (A - B) = \frac{tan (A) - tan (B)}{1 + tan (A) tan (B)}

$\tan (A - B) = \frac{\tan (A) - \tan (B)}{1 + \tan (A) \tan (B)}$ .

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

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

Step 3

Remove parentheses.

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

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

Step 4

Simplify the numerator.

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Step 4.1

The exact value of

tan (\frac{π}{3})

$\tan (\frac{π}{3})$ is

\sqrt{3}

$\sqrt{3}$ .

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

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

Step 4.2

The exact value of

tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

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

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

Step 4.3

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

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

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

Step 5

Simplify the denominator.

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Step 5.1

The exact value of

tan (\frac{π}{3})

$\tan (\frac{π}{3})$ is

\sqrt{3}

$\sqrt{3}$ .

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

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

Step 5.2

The exact value of

tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

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

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

Step 5.3

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

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

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

Step 6

Multiply

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

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

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

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

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

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

Step 7

Combine fractions.

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Step 7.1

Multiply

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

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

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

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

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

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

Step 7.2

Expand the denominator using the FOIL method.

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

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

Step 7.3

Simplify.

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

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

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

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

Step 8

Simplify the numerator.

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Step 8.1

Factor

- 1

$- 1$ out of

\sqrt{3}

$\sqrt{3}$ .

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

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

Step 8.2

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 8.3

Factor

- 1

$- 1$ out of

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

$- 1 (- \sqrt{3}) - 1 (1)$ .

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

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

Step 8.4

Reorder terms.

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

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

Step 8.5

Raise

1 - \sqrt{3}

$1 - \sqrt{3}$ to the power of

1

$1$ .

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

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

Step 8.6

Raise

1 - \sqrt{3}

$1 - \sqrt{3}$ to the power of

1

$1$ .

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

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

Step 8.7

Use the power rule

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

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

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

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

Step 8.8

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 9

Rewrite

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

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

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

$(1 - \sqrt{3}) (1 - \sqrt{3})$ .

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

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

Step 10

Expand

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

$(1 - \sqrt{3}) (1 - \sqrt{3})$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

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

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

Step 10.2

Apply the distributive property.

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

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

Step 10.3

Apply the distributive property.

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

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

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

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

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply

1

$1$ by

1

$1$ .

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

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

Step 11.1.2

Multiply

- \sqrt{3}

$- \sqrt{3}$ by

1

$1$ .

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

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

Step 11.1.3

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 11.1.4

Multiply

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

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

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Step 11.1.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 11.1.4.2

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

Step 11.1.4.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 11.1.4.4

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 11.1.4.5

Use the power rule

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

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

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

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

Step 11.1.4.6

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 11.1.5

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 11.1.5.1

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

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

Step 11.1.5.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 11.1.5.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 11.1.5.4

Cancel the common factor of

2

$2$ .

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Step 11.1.5.4.1

Cancel the common factor.

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

Step 11.1.5.4.2

Rewrite the expression.

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

Step 11.1.5.5

Evaluate the exponent.

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

Step 11.2

Add

$1$ and

$3$ .

$\frac{- 1 (4 - \sqrt{3} - \sqrt{3})}{- 2}$

Step 11.3

Subtract

$\sqrt{3}$ from

$- \sqrt{3}$ .

$\frac{- 1 (4 - 2 \sqrt{3})}{- 2}$

Step 12

Cancel the common factor of

$4 - 2 \sqrt{3}$ and

$- 2$ .

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Step 12.1

Factor

$2$ out of

$- 1 (4 - 2 \sqrt{3})$ .

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

Step 12.2

Move the negative one from the denominator of

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

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

Step 13

Rewrite

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

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

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

Step 14

Apply the distributive property.

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

Step 15

Multiply

$- 1$ by

$2$ .

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

Step 16

Multiply

$- 1 (- \sqrt{3})$ .

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Step 16.1

Multiply

$- 1$ by

$- 1$ .

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

Step 16.2

Multiply

$\sqrt{3}$ by

$1$ .

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

Step 17

Apply the distributive property.

$- - 2 - \sqrt{3}$

Step 18

Multiply

$- 1$ by

$- 2$ .

$2 - \sqrt{3}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$2 - \sqrt{3}$

Decimal Form:

$0.26794919 \dots$