Trigonometry Examples

$\cot (\frac{x}{2}) = 0$

Step 1

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

$\frac{x}{2} = arccot (0)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{2} = \frac{π}{2}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = π$

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.

$\frac{x}{2} = π + \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π + \frac{π}{2})$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π + \frac{π}{2})$

Step 5.2.1.1.2

Rewrite the expression.

$x = 2 (π + \frac{π}{2})$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $2 (π + \frac{π}{2})$ .

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

Simplify terms.

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Step 5.2.2.1.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 5.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 2 + π}{2}$

Step 5.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 5.2.2.1.2.3.1

Cancel the common factor.

$x = 2 \frac{π \cdot 2 + π}{2}$

Step 5.2.2.1.2.3.2

Rewrite the expression.

$x = π \cdot 2 + π$

Step 5.2.2.1.3

Move $2$ to the left of $π$ .

$x = 2 π + π$

Step 5.2.2.1.4

Add $2 π$ and $π$ .

$x = 3 π$

Step 6

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

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Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

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

$\frac{π}{| \frac{1}{2} |}$

Step 6.3

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

$\frac{π}{\frac{1}{2}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 6.5

Move $2$ to the left of $π$ .

$2 π$

Step 7

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

$x = π + 2 π n, 3 π + 2 π n$ , for any integer $n$

Step 8

Consolidate the answers.

$x = π + 2 π n$ , for any integer $n$