Trigonometry Examples

$\tan (105)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $105$ can be split into $45 + 60$ .

$\tan (45 + 60)$

Step 2

Use the sum formula for tangent to simplify the expression. The formula states that $\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

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

Step 3

Simplify the numerator.

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Step 3.1

The exact value of $\tan (45)$ is $1$ .

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

Step 3.2

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

Step 4

Simplify the denominator.

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Step 4.1

The exact value of $\tan (45)$ is $1$ .

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

Step 4.2

Multiply $- 1$ by $1$ .

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

Step 4.3

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

Step 4.4

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 5

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

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

Expand the denominator using the FOIL method.

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

Step 6.3

Simplify.

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.1.4

Combine using the product rule for radicals.

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

Step 10.1.5

Multiply $3$ by $3$ .

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

Step 10.1.6

Rewrite $9$ as $3^{2}$ .

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

Step 10.1.7

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

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

Step 10.2

Add $1$ and $3$ .

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

Step 10.3

Add $\sqrt{3}$ and $\sqrt{3}$ .

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

Step 11

Cancel the common factor of $4 + 2 \sqrt{3}$ and $- 2$ .

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Step 11.1

Factor $2$ out of $4$ .

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

Step 11.2

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

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

Step 11.3

Factor $2$ out of $2 (2) + 2 (\sqrt{3})$ .

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

Step 11.4

Move the negative one from the denominator of $\frac{2 + \sqrt{3}}{- 1}$ .

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

Step 12

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

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

Step 13

Apply the distributive property.

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

Step 14

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$