Trigonometry Examples

$\cot (\frac{x}{2}) = - 1$

Step 1

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

$\frac{x}{2} = arccot (- 1)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{2} = \frac{3 π}{4}$

Step 3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{3 π}{4}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{3 π}{4}$

Step 4.1.1.2

Rewrite the expression.

$x = 2 \frac{3 π}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{3 π}{2 (2)}$

Step 4.2.1.2

Cancel the common factor.

$x = 2 \frac{3 π}{2 \cdot 2}$

Step 4.2.1.3

Rewrite the expression.

$x = \frac{3 π}{2}$

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.

$\frac{x}{2} = \frac{3 π}{4} - π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$\frac{x}{2} = \frac{3 π}{4} - π + 2 π$

Step 6.2

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

$\frac{x}{2} = \frac{7 π}{4}$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{7 π}{4}$

Step 6.3.2

Simplify both sides of the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{7 π}{4}$

Step 6.3.2.1.1.2

Rewrite the expression.

$x = 2 \frac{7 π}{4}$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{7 π}{2 (2)}$

Step 6.3.2.2.1.2

Cancel the common factor.

$x = 2 \frac{7 π}{2 \cdot 2}$

Step 6.3.2.2.1.3

Rewrite the expression.

$x = \frac{7 π}{2}$

Step 7

Find the period of $\cot (\frac{x}{2})$ .

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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 $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 7.5

Move $2$ to the left of $π$ .

$2 π$

Step 8

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

$x = \frac{3 π}{2} + 2 π n, \frac{7 π}{2} + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{3 π}{2} + 2 π n$ , for any integer $n$