Algebra Examples

$\tan (\frac{π}{12}) = \cot (x - \frac{π}{36})$

Step 1

Rewrite the equation as $\cot (x - \frac{π}{36}) = \tan (\frac{π}{12})$ .

$\cot (x - \frac{π}{36}) = \tan (\frac{π}{12})$

Step 2

The exact value of $\tan (\frac{π}{12})$ is $2 - \sqrt{3}$ .

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Step 2.1

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

$\cot (x - \frac{π}{36}) = \tan (\frac{π}{4} - \frac{π}{6})$

Step 2.2

Apply the difference of angles identity.

$\cot (x - \frac{π}{36}) = \frac{\tan (\frac{π}{4}) - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 2.3

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

$\cot (x - \frac{π}{36}) = \frac{1 - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 2.4

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

$\cot (x - \frac{π}{36}) = \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 2.5

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

$\cot (x - \frac{π}{36}) = \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (\frac{π}{6})}$

Step 2.6

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

$\cot (x - \frac{π}{36}) = \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$

Step 2.7

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

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Step 2.7.1

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

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Step 2.7.1.1

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

$\cot (x - \frac{π}{36}) = \frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$

Step 2.7.1.2

Combine.

$\cot (x - \frac{π}{36}) = \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})}$

Step 2.7.2

Apply the distributive property.

$\cot (x - \frac{π}{36}) = \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}$

Step 2.7.3

Cancel the common factor of $3$ .

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Step 2.7.3.1

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

$\cot (x - \frac{π}{36}) = \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}$

Step 2.7.3.2

Cancel the common factor.

$\cot (x - \frac{π}{36}) = \frac{3 \cdot 1 + 3 \frac{- \sqrt{3}}{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}$

Step 2.7.3.3

Rewrite the expression.

$\cot (x - \frac{π}{36}) = \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 \frac{\sqrt{3}}{3})}$

Step 2.7.4

Multiply $3$ by $1$ .

$\cot (x - \frac{π}{36}) = \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot 1 \frac{\sqrt{3}}{3}}$

Step 2.7.5

Simplify the denominator.

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Step 2.7.5.1

Multiply $3$ by $1$ .

$\cot (x - \frac{π}{36}) = \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}$

Step 2.7.5.2

Cancel the common factor of $3$ .

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Step 2.7.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\cot (x - \frac{π}{36}) = \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}}$

Step 2.7.5.2.2

Cancel the common factor.

$\cot (x - \frac{π}{36}) = \frac{3 - \sqrt{3}}{3 + 3 \cdot 1 \frac{\sqrt{3}}{3}}$

Step 2.7.5.2.3

Rewrite the expression.

$\cot (x - \frac{π}{36}) = \frac{3 - \sqrt{3}}{3 + \sqrt{3}}$

Step 2.7.6

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

$\cot (x - \frac{π}{36}) = \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}$

Step 2.7.7

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

$\cot (x - \frac{π}{36}) = \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}$

Step 2.7.8

Expand the denominator using the FOIL method.

$\cot (x - \frac{π}{36}) = \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}$

Step 2.7.9

Simplify.

$\cot (x - \frac{π}{36}) = \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 2.7.10

Simplify the numerator.

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Step 2.7.10.1

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

$\cot (x - \frac{π}{36}) = \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}$

Step 2.7.10.2

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

$\cot (x - \frac{π}{36}) = \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}$

Step 2.7.10.3

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

$\cot (x - \frac{π}{36}) = \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}$

Step 2.7.10.4

Add $1$ and $1$ .

$\cot (x - \frac{π}{36}) = \frac{{(3 - \sqrt{3})}^{2}}{6}$

Step 2.7.11

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

$\cot (x - \frac{π}{36}) = \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 2.7.12

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

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Step 2.7.12.1

Apply the distributive property.

$\cot (x - \frac{π}{36}) = \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 2.7.12.2

Apply the distributive property.

$\cot (x - \frac{π}{36}) = \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 2.7.12.3

Apply the distributive property.

$\cot (x - \frac{π}{36}) = \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 2.7.13

Simplify and combine like terms.

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Step 2.7.13.1

Simplify each term.

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Step 2.7.13.1.1

Multiply $3$ by $3$ .

$\cot (x - \frac{π}{36}) = \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 2.7.13.1.2

Multiply $- 1$ by $3$ .

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 2.7.13.1.3

Multiply $3$ by $- 1$ .

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

Step 2.7.13.1.4

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

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Step 2.7.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

Step 2.7.13.1.4.2

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}$

Step 2.7.13.1.4.3

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

Step 2.7.13.1.4.4

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

Step 2.7.13.1.4.5

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Step 2.7.13.1.4.6

Add $1$ and $1$ .

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}$

Step 2.7.13.1.5

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

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Step 2.7.13.1.5.1

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}$

Step 2.7.13.1.5.2

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}$

Step 2.7.13.1.5.3

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

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Step 2.7.13.1.5.4

Cancel the common factor of $2$ .

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Step 2.7.13.1.5.4.1

Cancel the common factor.

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Step 2.7.13.1.5.4.2

Rewrite the expression.

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Step 2.7.13.1.5.5

Evaluate the exponent.

$\cot (x - \frac{π}{36}) = \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}$

Step 2.7.13.2

Add $9$ and $3$ .

$\cot (x - \frac{π}{36}) = \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}$

Step 2.7.13.3

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

$\cot (x - \frac{π}{36}) = \frac{12 - 6 \sqrt{3}}{6}$

Step 2.7.14

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

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Step 2.7.14.1

Factor $6$ out of $12$ .

$\cot (x - \frac{π}{36}) = \frac{6 \cdot 2 - 6 \sqrt{3}}{6}$

Step 2.7.14.2

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

$\cot (x - \frac{π}{36}) = \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}$

Step 2.7.14.3

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

$\cot (x - \frac{π}{36}) = \frac{6 (2 - \sqrt{3})}{6}$

Step 2.7.14.4

Cancel the common factors.

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Step 2.7.14.4.1

Factor $6$ out of $6$ .

$\cot (x - \frac{π}{36}) = \frac{6 (2 - \sqrt{3})}{6 (1)}$

Step 2.7.14.4.2

Cancel the common factor.

$\cot (x - \frac{π}{36}) = \frac{6 (2 - \sqrt{3})}{6 \cdot 1}$

Step 2.7.14.4.3

Rewrite the expression.

$\cot (x - \frac{π}{36}) = \frac{2 - \sqrt{3}}{1}$

Step 2.7.14.4.4

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

$\cot (x - \frac{π}{36}) = 2 - \sqrt{3}$

Step 3

Convert the right side of the equation to its decimal equivalent.

$\cot (x - \frac{π}{36}) = 0.26794919$

Step 4

Take the inverse cotangent of both sides of the equation to extract $x$ from inside the cotangent.

$x - \frac{π}{36} = arccot (0.26794919)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $arccot (0.26794919)$ .

$x - \frac{π}{36} = \frac{5 π}{12}$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Add $\frac{π}{36}$ to both sides of the equation.

$x = \frac{5 π}{12} + \frac{π}{36}$

Step 6.2

To write $\frac{5 π}{12}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{5 π}{12} \cdot \frac{3}{3} + \frac{π}{36}$

Step 6.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.1

Multiply $\frac{5 π}{12}$ by $\frac{3}{3}$ .

$x = \frac{5 π \cdot 3}{12 \cdot 3} + \frac{π}{36}$

Step 6.3.2

Multiply $12$ by $3$ .

$x = \frac{5 π \cdot 3}{36} + \frac{π}{36}$

Step 6.4

Combine the numerators over the common denominator.

$x = \frac{5 π \cdot 3 + π}{36}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $3$ by $5$ .

$x = \frac{15 π + π}{36}$

Step 6.5.2

Add $15 π$ and $π$ .

$x = \frac{16 π}{36}$

Step 6.6

Cancel the common factor of $16$ and $36$ .

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Step 6.6.1

Factor $4$ out of $16 π$ .

$x = \frac{4 (4 π)}{36}$

Step 6.6.2

Cancel the common factors.

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Step 6.6.2.1

Factor $4$ out of $36$ .

$x = \frac{4 (4 π)}{4 (9)}$

Step 6.6.2.2

Cancel the common factor.

$x = \frac{4 (4 π)}{4 \cdot 9}$

Step 6.6.2.3

Rewrite the expression.

$x = \frac{4 π}{9}$

Step 7

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x - \frac{π}{36} = π + \frac{5 π}{12}$

Step 8

Solve for $x$ .

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Step 8.1

Simplify $π + \frac{5 π}{12}$ .

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Step 8.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x - \frac{π}{36} = π \cdot \frac{12}{12} + \frac{5 π}{12}$

Step 8.1.2

Combine fractions.

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Step 8.1.2.1

Combine $π$ and $\frac{12}{12}$ .

$x - \frac{π}{36} = \frac{π \cdot 12}{12} + \frac{5 π}{12}$

Step 8.1.2.2

Combine the numerators over the common denominator.

$x - \frac{π}{36} = \frac{π \cdot 12 + 5 π}{12}$

Step 8.1.3

Simplify the numerator.

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Step 8.1.3.1

Move $12$ to the left of $π$ .

$x - \frac{π}{36} = \frac{12 \cdot π + 5 π}{12}$

Step 8.1.3.2

Add $12 π$ and $5 π$ .

$x - \frac{π}{36} = \frac{17 π}{12}$

Step 8.2

Move all terms not containing $x$ to the right side of the equation.

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Step 8.2.1

Add $\frac{π}{36}$ to both sides of the equation.

$x = \frac{17 π}{12} + \frac{π}{36}$

Step 8.2.2

To write $\frac{17 π}{12}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{17 π}{12} \cdot \frac{3}{3} + \frac{π}{36}$

Step 8.2.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.1

Multiply $\frac{17 π}{12}$ by $\frac{3}{3}$ .

$x = \frac{17 π \cdot 3}{12 \cdot 3} + \frac{π}{36}$

Step 8.2.3.2

Multiply $12$ by $3$ .

$x = \frac{17 π \cdot 3}{36} + \frac{π}{36}$

Step 8.2.4

Combine the numerators over the common denominator.

$x = \frac{17 π \cdot 3 + π}{36}$

Step 8.2.5

Simplify the numerator.

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Step 8.2.5.1

Multiply $3$ by $17$ .

$x = \frac{51 π + π}{36}$

Step 8.2.5.2

Add $51 π$ and $π$ .

$x = \frac{52 π}{36}$

Step 8.2.6

Cancel the common factor of $52$ and $36$ .

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Step 8.2.6.1

Factor $4$ out of $52 π$ .

$x = \frac{4 (13 π)}{36}$

Step 8.2.6.2

Cancel the common factors.

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Step 8.2.6.2.1

Factor $4$ out of $36$ .

$x = \frac{4 (13 π)}{4 (9)}$

Step 8.2.6.2.2

Cancel the common factor.

$x = \frac{4 (13 π)}{4 \cdot 9}$

Step 8.2.6.2.3

Rewrite the expression.

$x = \frac{13 π}{9}$

Step 9

Find the period of $\cot (x - \frac{π}{36})$ .

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Step 9.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 9.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

The period of the $\cot (x - \frac{π}{36})$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{4 π}{9} + π n, \frac{13 π}{9} + π n$ , for any integer $n$

Step 11

Consolidate the answers.

$x = \frac{4 π}{9} + π n$ , for any integer $n$