Trigonometry Examples

$\cot (x) = \frac{1}{\tan (\frac{7}{21})}$

Step 1

Simplify $\frac{1}{\tan (\frac{7}{21})}$ .

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Step 1.1

Cancel the common factor of $7$ and $21$ .

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Step 1.1.1

Factor $7$ out of $7$ .

$\cot (x) = \frac{1}{\tan (\frac{7 (1)}{21})}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $7$ out of $21$ .

$\cot (x) = \frac{1}{\tan (\frac{7 \cdot 1}{7 \cdot 3})}$

Step 1.1.2.2

Cancel the common factor.

$\cot (x) = \frac{1}{\tan (\frac{7 \cdot 1}{7 \cdot 3})}$

Step 1.1.2.3

Rewrite the expression.

$\cot (x) = \frac{1}{\tan (\frac{1}{3})}$

Step 1.2

Rewrite $\tan (\frac{1}{3})$ in terms of sines and cosines.

$\cot (x) = \frac{1}{\frac{\sin (\frac{1}{3})}{\cos (\frac{1}{3})}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (\frac{1}{3})}{\cos (\frac{1}{3})}$ .

$\cot (x) = \frac{\cos (\frac{1}{3})}{\sin (\frac{1}{3})}$

Step 1.4

Convert from $\frac{\cos (\frac{1}{3})}{\sin (\frac{1}{3})}$ to $\cot (\frac{1}{3})$ .

$\cot (x) = \cot (\frac{1}{3})$

Step 1.5

Evaluate $\cot (\frac{1}{3})$ .

$\cot (x) = 2.88805703$

Step 2

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

$x = arccot (2.88805703)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccot (2.88805703)$ .

$x = 0.\overline{3}$

Step 4

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.

$x = (3.14159265) + 0.\overline{3}$

Step 5

Solve for $x$ .

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Step 5.1

Remove parentheses.

$x = 3.14159265 + 0.\overline{3}$

Step 5.2

Remove parentheses.

$x = (3.14159265) + 0.\overline{3}$

Step 5.3

Add $3.14159265$ and $0.\overline{3}$ .

$x = 3.47492598$

Step 6

Find the period of $\cot (x)$ .

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Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

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

$\frac{π}{| 1 |}$

Step 6.3

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

$\frac{π}{1}$

Step 6.4

Divide $π$ by $1$ .

$π$

Step 7

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

$x = 0.\overline{3} + π n, 3.47492598 + π n$ , for any integer $n$

Step 8

Consolidate $0.\overline{3} + π n$ and $3.47492598 + π n$ to $0.\overline{3} + π n$ .

$x = 0.\overline{3} + π n$ , for any integer $n$