Trigonometry Examples

$\tan (\frac{13 π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{13 π}{12}$ can be split into $\frac{3 π}{4} + \frac{π}{3}$ .

$\tan (\frac{3 π}{4} + \frac{π}{3})$

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 3.4

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

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

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

Step 4.2

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

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

Step 4.3

Multiply $- (- 1 \cdot 1)$ .

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Step 4.3.1

Multiply $- 1$ by $1$ .

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

Step 4.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.4

Multiply $\tan (\frac{π}{3})$ by $1$ .

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

Step 4.5

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

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

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 7.2

Factor $- 1$ out of $\sqrt{3}$ .

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

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 $1$ .

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

Step 10.1.2

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

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

Step 10.1.3

Multiply $- 1$ by $1$ .

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

Step 10.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 10.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 10.1.4.2

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

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

Step 10.1.4.3

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

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

Step 10.1.4.4

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

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

Step 10.1.4.5

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

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

Step 10.1.4.6

Add $1$ and $1$ .

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

Step 10.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 10.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 10.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 10.1.5.4

Cancel the common factor of $2$ .

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Step 10.1.5.4.1

Cancel the common factor.

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

Step 10.1.5.4.2

Rewrite the expression.

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

Step 10.1.5.5

Evaluate the exponent.

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

Step 10.2

Add $1$ and $3$ .

$\frac{- 1 (4 - \sqrt{3} - \sqrt{3})}{- 2}$

Step 10.3

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$\frac{- 1 (4 - 2 \sqrt{3})}{- 2}$

Step 11

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

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Step 11.1

Factor $2$ out of $- 1 (4 - 2 \sqrt{3})$ .

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

Step 11.2

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

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

Step 12

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

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

Step 13

Apply the distributive property.

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

Step 14

Multiply $- 1$ by $2$ .

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

Step 15

Multiply $- 1 (- \sqrt{3})$ .

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Step 15.1

Multiply $- 1$ by $- 1$ .

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

Step 15.2

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

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

Step 16

Apply the distributive property.

$- - 2 - \sqrt{3}$

Step 17

Multiply $- 1$ by $- 2$ .

$2 - \sqrt{3}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$2 - \sqrt{3}$

Decimal Form:

$0.26794919 \dots$