Algebra Examples

$\cot (- \frac{7 π}{12})$

Step 1

Replace $\cot (- \frac{7 π}{12})$ with an equivalent expression $\frac{1}{\tan (- \frac{7 π}{12})}$ using the fundamental identities.

$\frac{1}{\tan (- \frac{7 π}{12})}$

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, $- \frac{7 π}{12}$ can be split into $\frac{π}{4} - \frac{5 π}{6}$ .

$\frac{1}{\tan (\frac{π}{4} - \frac{5 π}{6})}$

Step 2.2

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

$\frac{1}{\frac{\tan (\frac{π}{4}) - \tan (\frac{5 π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.3

Remove parentheses.

$\frac{1}{\frac{\tan (\frac{π}{4}) - \tan (\frac{5 π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

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

$\frac{1}{\frac{1 - \tan (\frac{5 π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

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.

$\frac{1}{\frac{1 + \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.4.3

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

$\frac{1}{\frac{1 + \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.4.4

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

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Step 2.4.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{\frac{1 + 1 (\frac{\sqrt{3}}{3})}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.4.4.2

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

$\frac{1}{\frac{1 + \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.4.5

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

$\frac{1}{\frac{\frac{3}{3} + \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.4.6

Combine the numerators over the common denominator.

$\frac{1}{\frac{\frac{3 + \sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{5 π}{6})}}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

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

$\frac{1}{\frac{\frac{3 + \sqrt{3}}{3}}{1 + 1 \tan (\frac{5 π}{6})}}$

Step 2.5.2

Multiply $\tan (\frac{5 π}{6})$ by $1$ .

$\frac{1}{\frac{\frac{3 + \sqrt{3}}{3}}{1 + \tan (\frac{5 π}{6})}}$

Step 2.5.3

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.

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

Combine the numerators over the common denominator.

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

Step 2.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.7

Cancel the common factor of $3$ .

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Step 2.7.1

Cancel the common factor.

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

Step 2.7.2

Rewrite the expression.

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

Step 2.8

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

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

Step 2.9

Simplify terms.

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Step 2.9.1

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

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

Step 2.9.2

Expand the denominator using the FOIL method.

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

Step 2.9.3

Simplify.

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

Step 2.9.4

Apply the distributive property.

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

Step 2.9.5

Cancel the common factor of $3$ .

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Step 2.9.5.1

Factor $3$ out of $6$ .

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

Step 2.9.5.2

Cancel the common factor.

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

Step 2.9.5.3

Rewrite the expression.

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

Step 2.9.6

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

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

Step 2.10

Simplify each term.

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

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

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

Step 2.10.3

Combine using the product rule for radicals.

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

Step 2.10.4

Simplify each term.

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Step 2.10.4.1

Multiply $3$ by $3$ .

$\frac{1}{\frac{3 + \sqrt{3}}{2} + \frac{3 \sqrt{3} + \sqrt{9}}{6}}$

Step 2.10.4.2

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

$\frac{1}{\frac{3 + \sqrt{3}}{2} + \frac{3 \sqrt{3} + \sqrt{3^{2}}}{6}}$

Step 2.10.4.3

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

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

Step 2.10.5

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

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Step 2.10.5.1

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

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

Step 2.10.5.2

Factor $3$ out of $3$ .

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

Step 2.10.5.3

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

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

Step 2.10.5.4

Cancel the common factors.

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Step 2.10.5.4.1

Factor $3$ out of $6$ .

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

Step 2.10.5.4.2

Cancel the common factor.

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

Step 2.10.5.4.3

Rewrite the expression.

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

Step 2.11

Simplify terms.

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Step 2.11.1

Combine the numerators over the common denominator.

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

Step 2.11.2

Add $3$ and $1$ .

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

Step 2.11.3

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

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

Step 2.11.4

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

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Step 2.11.4.1

Factor $2$ out of $4$ .

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

Step 2.11.4.2

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

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

Step 2.11.4.3

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

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

Step 2.11.4.4

Cancel the common factors.

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Step 2.11.4.4.1

Factor $2$ out of $2$ .

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

Step 2.11.4.4.2

Cancel the common factor.

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

Step 2.11.4.4.3

Rewrite the expression.

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

Step 2.11.4.4.4

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

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

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.

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

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:

$0.26794919 \dots$