Algebra Examples

$\cot (x) = 1$ , $π \leq x \leq \frac{3 π}{2}$

Step 1

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

$x = arccot (1)$

Step 2

Simplify the right side.

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Step 2.1

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

$x = \frac{π}{4}$

Step 3

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.

$x = π + \frac{π}{4}$

Step 4

Simplify $π + \frac{π}{4}$ .

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Step 4.1

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

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

Step 4.2

Combine fractions.

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Step 4.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 4.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 5

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

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Step 5.1

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

$\frac{π}{| b |}$

Step 5.2

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

$\frac{π}{| 1 |}$

Step 5.3

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

$\frac{π}{1}$

Step 5.4

Divide $π$ by $1$ .

$π$

Step 6

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

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

Step 7

Consolidate the answers.

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

Step 8

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[π, \frac{3 π}{2}]$ .

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Step 8.1

Plug in $1$ for $n$ .

$\frac{π}{4} + π (1)$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $π$ by $1$ .

$\frac{π}{4} + π$

Step 8.2.2

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

$\frac{π}{4} + π \cdot \frac{4}{4}$

Step 8.2.3

Combine fractions.

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Step 8.2.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π}{4} + \frac{π \cdot 4}{4}$

Step 8.2.3.2

Combine the numerators over the common denominator.

$\frac{π + π \cdot 4}{4}$

Step 8.2.4

Simplify the numerator.

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Step 8.2.4.1

Move $4$ to the left of $π$ .

$\frac{π + 4 \cdot π}{4}$

Step 8.2.4.2

Add $π$ and $4 π$ .

$\frac{5 π}{4}$

Step 8.3

The interval $[π, \frac{3 π}{2}]$ contains $\frac{5 π}{4}$ .

$x = \frac{5 π}{4}$