Trigonometry Examples

$\tan (\frac{5 π}{12})$

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{5 π}{12}$ can be split into $\frac{π}{6} + \frac{π}{4}$ .

$\tan (\frac{π}{6} + \frac{π}{4})$

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

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

Simplify the numerator.

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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 $1$ .

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Simplify the denominator.

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

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

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

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

Multiply $- 1$ by $1$ .

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify terms.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Apply the distributive property.

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

Combine $\sqrt{3}$ and $\frac{3 + \sqrt{3}}{6}$ .

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

Cancel the common factor of $3$ .

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Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify each term.

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Apply the distributive property.

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

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

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

Combine using the product rule for radicals.

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

Simplify each term.

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Multiply $3$ by $3$ .

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

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

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

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

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

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

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

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

Factor $3$ out of $3$ .

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

Factor $3$ out of $3 (\sqrt{3}) + 3 (1)$ .

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

Cancel the common factors.

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Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify terms.

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Combine the numerators over the common denominator.

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

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

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

Add $1$ and $3$ .

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

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

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Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

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:

$3.73205080 \dots$