Find the Exact Value tan((5pi)/12)

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Trigonometry

$\tan (\frac{5 π}{12})$

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\tan (\frac{π}{6} + \frac{π}{4})$

Apply the sum of angles identity.

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

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

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

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Multiply the numerator and denominator of the complex fraction by $3$ .

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $1$ .

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

Simplify the denominator.

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Multiply $3$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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Reorder terms.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $3$ .

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

Move $3$ to the left of $\sqrt{3}$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

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

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

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

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

Add $9$ and $3$ .

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

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

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

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Factor $6$ out of $12$ .

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

Factor $6$ out of $6 \sqrt{3}$ .

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

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

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

Cancel the common factors.

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Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 + \sqrt{3}$ by $1$ .

$2 + \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$

Do you know how to Find the Exact Value tan((5pi)/12)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name

Name	one billion two hundred thirty-seven million seven hundred forty-four thousand three hundred ninety-eight

Interesting facts

1237744398 has 32 divisors, whose sum is 2132036640
The reverse of 1237744398 is 8934477321
Previous prime number is 19

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 48
Digital Root 3

Name

Name	one billion six hundred twenty-two million six hundred ninety-three thousand thirty-one

Interesting facts

1622693031 has 4 divisors, whose sum is 2163590712
The reverse of 1622693031 is 1303962261
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 33
Digital Root 6

Name

Name	seventy million three hundred eight thousand seven hundred nineteen

Interesting facts

70308719 has 16 divisors, whose sum is 81158112
The reverse of 70308719 is 91780307
Previous prime number is 83

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 35
Digital Root 8

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