Trigonometry Examples

$\cot (x + π) = 0$

Step 1

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

$x + π = arccot (0)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$x + π = \frac{π}{2}$

Step 3

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

Tap for more steps...

Step 3.1

Subtract $π$ from both sides of the equation.

$x = \frac{π}{2} - π$

Step 3.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{π}{2} - π \cdot \frac{2}{2}$

Step 3.3

Combine $- π$ and $\frac{2}{2}$ .

$x = \frac{π}{2} + \frac{- π \cdot 2}{2}$

Step 3.4

Combine the numerators over the common denominator.

$x = \frac{π - π \cdot 2}{2}$

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Multiply $2$ by $- 1$ .

$x = \frac{π - 2 π}{2}$

Step 3.5.2

Subtract $2 π$ from $π$ .

$x = \frac{- π}{2}$

Step 3.6

Move the negative in front of the fraction.

$x = - \frac{π}{2}$

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.

$x + π = π + \frac{π}{2}$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Simplify $π + \frac{π}{2}$ .

Tap for more steps...

Step 5.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x + π = π \cdot \frac{2}{2} + \frac{π}{2}$

Step 5.1.2

Combine fractions.

Tap for more steps...

Step 5.1.2.1

Combine $π$ and $\frac{2}{2}$ .

$x + π = \frac{π \cdot 2}{2} + \frac{π}{2}$

Step 5.1.2.2

Combine the numerators over the common denominator.

$x + π = \frac{π \cdot 2 + π}{2}$

Step 5.1.3

Simplify the numerator.

Tap for more steps...

Step 5.1.3.1

Move $2$ to the left of $π$ .

$x + π = \frac{2 \cdot π + π}{2}$

Step 5.1.3.2

Add $2 π$ and $π$ .

$x + π = \frac{3 π}{2}$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $π$ from both sides of the equation.

$x = \frac{3 π}{2} - π$

Step 5.2.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{3 π}{2} - π \cdot \frac{2}{2}$

Step 5.2.3

Combine $- π$ and $\frac{2}{2}$ .

$x = \frac{3 π}{2} + \frac{- π \cdot 2}{2}$

Step 5.2.4

Combine the numerators over the common denominator.

$x = \frac{3 π - π \cdot 2}{2}$

Step 5.2.5

Simplify the numerator.

Tap for more steps...

Step 5.2.5.1

Multiply $2$ by $- 1$ .

$x = \frac{3 π - 2 π}{2}$

Step 5.2.5.2

Subtract $2 π$ from $3 π$ .

$x = \frac{π}{2}$

Step 6

Find the period of $\cot (x + π)$ .

Tap for more steps...

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

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

Tap for more steps...

Step 7.1

Add $π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + π$

Step 7.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$π \cdot \frac{2}{2} - \frac{π}{2}$

Step 7.3

Combine fractions.

Tap for more steps...

Step 7.3.1

Combine $π$ and $\frac{2}{2}$ .

$\frac{π \cdot 2}{2} - \frac{π}{2}$

Step 7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 2 - π}{2}$

Step 7.4

Simplify the numerator.

Tap for more steps...

Step 7.4.1

Move $2$ to the left of $π$ .

$\frac{2 \cdot π - π}{2}$

Step 7.4.2

Subtract $π$ from $2 π$ .

$\frac{π}{2}$

Step 7.5

List the new angles.

$x = \frac{π}{2}$

Step 8

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

$x = \frac{π}{2} + π n, \frac{π}{2} + π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{2} + π n$ , for any integer $n$