Trigonometry Examples

$3 \cot (x + 2) = 5$

Step 1

Divide each term in $3 \cot (x + 2) = 5$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 \cot (x + 2) = 5$ by $3$ .

$\frac{3 \cot (x + 2)}{3} = \frac{5}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 \cot (x + 2)}{3} = \frac{5}{3}$

Step 1.2.1.2

Divide $\cot (x + 2)$ by $1$ .

$\cot (x + 2) = \frac{5}{3}$

Step 2

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

$x + 2 = arccot (\frac{5}{3})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccot (\frac{5}{3})$ .

$x + 2 = 0.5404195$

Step 4

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

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Step 4.1

Subtract $2$ from both sides of the equation.

$x = 0.5404195 - 2$

Step 4.2

Subtract $2$ from $0.5404195$ .

$x = - 1.45958049$

Step 5

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.

$x + 2 = (3.14159265) + 0.5404195$

Step 6

Solve for $x$ .

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Step 6.1

Add $3.14159265$ and $0.5404195$ .

$x + 2 = 3.68201215$

Step 6.2

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

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Step 6.2.1

Subtract $2$ from both sides of the equation.

$x = 3.68201215 - 2$

Step 6.2.2

Subtract $2$ from $3.68201215$ .

$x = 1.68201215$

Step 7

Find the period of $\cot (x + 2)$ .

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

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

$\frac{π}{| 1 |}$

Step 7.3

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

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

Add $π$ to every negative angle to get positive angles.

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Step 8.1

Add $π$ to $- 1.45958049$ to find the positive angle.

$- 1.45958049 + π$

Step 8.2

Replace with decimal approximation.

$3.14159265 - 1.45958049$

Step 8.3

Subtract $1.45958049$ from $3.14159265$ .

$1.68201215$

Step 8.4

List the new angles.

$x = 1.68201215$

Step 9

The period of the $\cot (x + 2)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 1.68201215 + π n, 1.68201215 + π n$ , for any integer $n$

Step 10

Consolidate $1.68201215 + π n$ and $1.68201215 + π n$ to $1.68201215 + π n$ .

$x = 1.68201215 + π n$ , for any integer $n$