Trigonometry Examples

$4 + 2 \cot (x) = 12 - \frac{\sqrt{3}}{3} + \cot (x)$

Step 1

Move all terms containing $\cot (x)$ to the left side of the equation.

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Step 1.1

Subtract $\cot (x)$ from both sides of the equation.

$4 + 2 \cot (x) - \cot (x) = 12 - \frac{\sqrt{3}}{3}$

Step 1.2

Subtract $\cot (x)$ from $2 \cot (x)$ .

$4 + \cot (x) = 12 - \frac{\sqrt{3}}{3}$

Step 2

Move all terms not containing $\cot (x)$ to the right side of the equation.

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Step 2.1

Subtract $4$ from both sides of the equation.

$\cot (x) = 12 - \frac{\sqrt{3}}{3} - 4$

Step 2.2

Subtract $4$ from $12$ .

$\cot (x) = 8 - \frac{\sqrt{3}}{3}$

Step 3

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

$\cot (x) = 7.42264973$

Step 4

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

$x = arccot (7.42264973)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $arccot (7.42264973)$ .

$x = 0.13391645$

Step 6

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.

$x = (3.14159265) + 0.13391645$

Step 7

Solve for $x$ .

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Step 7.1

Remove parentheses.

$x = 3.14159265 + 0.13391645$

Step 7.2

Remove parentheses.

$x = (3.14159265) + 0.13391645$

Step 7.3

Add $3.14159265$ and $0.13391645$ .

$x = 3.27550911$

Step 8

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

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| 1 |}$

Step 8.3

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

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

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

$x = 0.13391645 + π n, 3.27550911 + π n$ , for any integer $n$

Step 10

Consolidate $0.13391645 + π n$ and $3.27550911 + π n$ to $0.13391645 + π n$ .

$x = 0.13391645 + π n$ , for any integer $n$