Find the Exact Value tan(105)

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Trigonometry

$\tan (105)$

Split $105$ into two angles where the values of the six trigonometric functions are known.

$\tan (45 + 60)$

Apply the sum of angles identity.

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

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

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

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

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

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

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

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

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

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

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Simplify the denominator.

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Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

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Factor $2$ out of $4$ .

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

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

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

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

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$

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Name

Name	three hundred thirty-three million one hundred seventy-nine thousand two hundred fifty-three

Interesting facts

333179253 has 16 divisors, whose sum is 411480000
The reverse of 333179253 is 352971333
Previous prime number is 53

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 36
Digital Root 9

Name

Name	seven hundred fifty-three million four hundred seventy-three thousand nine hundred sixty-eight

Interesting facts

753473968 has 128 divisors, whose sum is 4125201696
The reverse of 753473968 is 869374357
Previous prime number is 241

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 52
Digital Root 7

Name

Name	two hundred twenty-six million nine hundred eighty-six thousand nine hundred seventy-two

Interesting facts

226986972 has 16 divisors, whose sum is 680960952
The reverse of 226986972 is 279689622
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 51
Digital Root 6

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