Trigonometry Examples

$5 \cot (x) + 4 = - 1$

Step 1

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

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Step 1.1

Subtract $4$ from both sides of the equation.

$5 \cot (x) = - 1 - 4$

Step 1.2

Subtract $4$ from $- 1$ .

$5 \cot (x) = - 5$

Step 2

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

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Step 2.1

Divide each term in $5 \cot (x) = - 5$ by $5$ .

$\frac{5 \cot (x)}{5} = \frac{- 5}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 \cot (x)}{5} = \frac{- 5}{5}$

Step 2.2.1.2

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

$\cot (x) = \frac{- 5}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 5$ by $5$ .

$\cot (x) = - 1$

Step 3

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

$x = arccot (- 1)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $arccot (- 1)$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 5

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

$x = \frac{3 π}{4} - π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$x = \frac{3 π}{4} - π + 2 π$

Step 6.2

The resulting angle of $\frac{7 π}{4}$ is positive and coterminal with $\frac{3 π}{4} - π$ .

$x = \frac{7 π}{4}$

Step 7

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

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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

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

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{3 π}{4} + π n$ , for any integer $n$