Algebra Examples

$\cot (- \frac{5 π}{12})$

Step 1

Replace $\cot (- \frac{5 π}{12})$ with an equivalent expression $\frac{1}{\tan (- \frac{5 π}{12})}$ using the fundamental identities.

$\frac{1}{\tan (- \frac{5 π}{12})}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

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 $\frac{π}{3} - \frac{3 π}{4}$ .

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

Step 2.2

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

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

Step 2.3

Remove parentheses.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

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

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

Step 2.4.2

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.

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

Step 2.4.3

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

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

Step 2.4.4

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

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Step 2.4.4.1

Multiply $- 1$ by $1$ .

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

Step 2.4.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.5

Simplify the denominator.

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Step 2.5.1

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

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

Step 2.5.2

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.

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

Step 2.5.3

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

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

Step 2.5.4

Multiply $- 1$ by $1$ .

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

Step 2.5.5

Move $- 1$ to the left of $\sqrt{3}$ .

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

Step 2.5.6

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

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

Step 2.6

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

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

Step 2.7

Combine fractions.

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Step 2.7.1

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

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

Step 2.7.2

Expand the denominator using the FOIL method.

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

Step 2.7.3

Simplify.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Reorder terms.

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

Step 2.8.2

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

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

Step 2.8.3

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

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

Step 2.8.4

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

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

Step 2.8.5

Add $1$ and $1$ .

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

Step 2.9

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

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

Step 2.10

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

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

Apply the distributive property.

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

Step 2.10.3

Apply the distributive property.

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

Step 2.11

Simplify and combine like terms.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

Multiply $1$ by $1$ .

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

Step 2.11.1.2

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

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

Step 2.11.1.3

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

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

Step 2.11.1.4

Combine using the product rule for radicals.

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

Step 2.11.1.5

Multiply $3$ by $3$ .

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

Step 2.11.1.6

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

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

Step 2.11.1.7

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

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

Step 2.11.2

Add $1$ and $3$ .

$\frac{1}{\frac{4 + \sqrt{3} + \sqrt{3}}{- 2}}$

Step 2.11.3

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

$\frac{1}{\frac{4 + 2 \sqrt{3}}{- 2}}$

Step 2.12

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

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Step 2.12.1

Factor $2$ out of $4$ .

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

Step 2.12.2

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

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

Step 2.12.3

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

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

Step 2.12.4

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

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

Step 2.13

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

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

Step 2.14

Apply the distributive property.

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

Step 2.15

Multiply $- 1$ by $2$ .

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

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{- 2 - \sqrt{3}}$ by $\frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$ .

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

Step 3.2

Multiply $\frac{1}{- 2 - \sqrt{3}}$ by $\frac{- 2 + \sqrt{3}}{- 2 + \sqrt{3}}$ .

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

Step 3.3

Expand the denominator using the FOIL method.

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

Step 3.4

Simplify.

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

Step 3.5

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

$- 2 + \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$