Trigonometry Examples

$2 \cot (x + 4) = 2$

Step 1

Divide each term in $2 \cot (x + 4) = 2$ by $2$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $2 \cot (x + 4) = 2$ by $2$ .

$\frac{2 \cot (x + 4)}{2} = \frac{2}{2}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cot (x + 4)}{2} = \frac{2}{2}$

Step 1.2.1.2

Divide $\cot (x + 4)$ by $1$ .

$\cot (x + 4) = \frac{2}{2}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Divide $2$ by $2$ .

$\cot (x + 4) = 1$

Step 2

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

$x + 4 = arccot (1)$

Step 3

Simplify the right side.

Tap for more steps...

Step 3.1

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

$x + 4 = \frac{π}{4}$

Step 4

Subtract $4$ from both sides of the equation.

$x = \frac{π}{4} - 4$

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.

$x + 4 = π + \frac{π}{4}$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

Step 6.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x + 4 = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 6.1.2

Combine fractions.

Tap for more steps...

Step 6.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$x + 4 = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 6.1.2.2

Combine the numerators over the common denominator.

$x + 4 = \frac{π \cdot 4 + π}{4}$

Step 6.1.3

Simplify the numerator.

Tap for more steps...

Step 6.1.3.1

Move $4$ to the left of $π$ .

$x + 4 = \frac{4 \cdot π + π}{4}$

Step 6.1.3.2

Add $4 π$ and $π$ .

$x + 4 = \frac{5 π}{4}$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = \frac{5 π}{4} - 4$

Step 7

Find the period of $\cot (x + 4)$ .

Tap for more steps...

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 $\cot (x + 4)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{4} - 4 + π n, \frac{5 π}{4} - 4 + π n$ , for any integer $n$

Step 9

Consolidate the answers.

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