Trigonometry Examples

tan (105)

$\tan (105)$

Step 1

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

105

$105$ can be split into

45 + 60

$45 + 60$ .

tan (45 + 60)

$\tan (45 + 60)$

Step 2

Use the sum 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 (45) + tan (60)}{1 - tan (45) tan (60)}

$\frac{\tan (45) + \tan (60)}{1 - \tan (45) \tan (60)}$

Step 3

Simplify the numerator.

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Step 3.1

The exact value of

tan (45)

$\tan (45)$ is

1

$1$ .

\frac{1 + tan (60)}{1 - tan (45) tan (60)}

$\frac{1 + \tan (60)}{1 - \tan (45) \tan (60)}$

Step 3.2

The exact value of

tan (60)

$\tan (60)$ is

\sqrt{3}

$\sqrt{3}$ .

\frac{1 + \sqrt{3}}{1 - tan (45) tan (60)}

$\frac{1 + \sqrt{3}}{1 - \tan (45) \tan (60)}$

\frac{1 + \sqrt{3}}{1 - tan (45) tan (60)}

$\frac{1 + \sqrt{3}}{1 - \tan (45) \tan (60)}$

Step 4

Simplify the denominator.

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Step 4.1

The exact value of

tan (45)

$\tan (45)$ is

1

$1$ .

\frac{1 + \sqrt{3}}{1 - 1 \cdot 1 tan (60)}

$\frac{1 + \sqrt{3}}{1 - 1 \cdot 1 \tan (60)}$

Step 4.2

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{1 + \sqrt{3}}{1 - 1 tan (60)}

$\frac{1 + \sqrt{3}}{1 - 1 \tan (60)}$

Step 4.3

The exact value of

tan (60)

$\tan (60)$ is

\sqrt{3}

$\sqrt{3}$ .

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

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

Step 4.4

Rewrite

- 1 \sqrt{3}

$- 1 \sqrt{3}$ as

- \sqrt{3}

$- \sqrt{3}$ .

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

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

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

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

Step 5

Multiply

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

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

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

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

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

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

Step 6

Combine fractions.

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Step 6.1

Multiply

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

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

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

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

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

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

Step 6.2

Expand the denominator using the FOIL method.

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

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

Step 6.3

Simplify.

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

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

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

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

Step 7

Simplify the numerator.

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Step 7.1

Raise

1 + \sqrt{3}

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

1

$1$ .

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

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

Step 7.2

Raise

1 + \sqrt{3}

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

1

$1$ .

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

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

Step 7.3

Use the power rule

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

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

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

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

Step 7.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 8

Rewrite

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

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

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

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

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

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

Step 9

Expand

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

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

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Step 9.1

Apply the distributive property.

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

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

Step 9.2

Apply the distributive property.

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

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

Step 9.3

Apply the distributive property.

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

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

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

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

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply

1

$1$ 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 10.1.2

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

Step 10.1.3

Multiply

\sqrt{3}

$\sqrt{3}$ by

1

$1$ .

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

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

Step 10.1.4

Combine using the product rule for radicals.

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

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

Step 10.1.5

Multiply

3

$3$ by

3

$3$ .

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

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

Step 10.1.6

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

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

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

Step 10.1.7

Pull terms out from under the radical, assuming positive real numbers.

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

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

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

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

Step 10.2

Add

1

$1$ and

3

$3$ .

\frac{4 + \sqrt{3} + \sqrt{3}}{- 2}

$\frac{4 + \sqrt{3} + \sqrt{3}}{- 2}$

Step 10.3

Add

\sqrt{3}

$\sqrt{3}$ and

\sqrt{3}

$\sqrt{3}$ .

\frac{4 + 2 \sqrt{3}}{- 2}

$\frac{4 + 2 \sqrt{3}}{- 2}$

\frac{4 + 2 \sqrt{3}}{- 2}

$\frac{4 + 2 \sqrt{3}}{- 2}$

Step 11

Cancel the common factor of

4 + 2 \sqrt{3}

$4 + 2 \sqrt{3}$ and

- 2

$- 2$ .

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Step 11.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 11.2

Factor

2

$2$ out of

2 \sqrt{3}

$2 \sqrt{3}$ .

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

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

Step 11.3

Factor

2

$2$ out of

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

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

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

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

Step 11.4

Move the negative one from the denominator of

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

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

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

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

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

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

Step 12

Rewrite

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

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

- (2 + \sqrt{3})

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

- (2 + \sqrt{3})

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

Step 13

Apply the distributive property.

- 1 \cdot 2 - \sqrt{3}

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

Step 14

Multiply

- 1

$- 1$ by

2

$2$ .

- 2 - \sqrt{3}

$- 2 - \sqrt{3}$

Step 15

The result can be shown in multiple forms.

Exact Form:

- 2 - \sqrt{3}

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$