Trigonometry Examples

$\cot (2 θ - \frac{π}{2}) = 1$

Step 1

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

$2 θ - \frac{π}{2} = arccot (1)$

Step 2

Simplify the right side.

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Step 2.1

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

$2 θ - \frac{π}{2} = \frac{π}{4}$

Step 3

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

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Step 3.1

Add $\frac{π}{2}$ to both sides of the equation.

$2 θ = \frac{π}{4} + \frac{π}{2}$

Step 3.2

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

$2 θ = \frac{π}{4} + \frac{π}{2} \cdot \frac{2}{2}$

Step 3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$2 θ = \frac{π}{4} + \frac{π \cdot 2}{2 \cdot 2}$

Step 3.3.2

Multiply $2$ by $2$ .

$2 θ = \frac{π}{4} + \frac{π \cdot 2}{4}$

Step 3.4

Combine the numerators over the common denominator.

$2 θ = \frac{π + π \cdot 2}{4}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Move $2$ to the left of $π$ .

$2 θ = \frac{π + 2 \cdot π}{4}$

Step 3.5.2

Add $π$ and $2 π$ .

$2 θ = \frac{3 π}{4}$

Step 4

Divide each term in $2 θ = \frac{3 π}{4}$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 θ = \frac{3 π}{4}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{3 π}{4}}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{3 π}{4}}{2}$

Step 4.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{3 π}{4}}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{3 π}{4} \cdot \frac{1}{2}$

Step 4.3.2

Multiply $\frac{3 π}{4} \cdot \frac{1}{2}$ .

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Step 4.3.2.1

Multiply $\frac{3 π}{4}$ by $\frac{1}{2}$ .

$θ = \frac{3 π}{4 \cdot 2}$

Step 4.3.2.2

Multiply $4$ by $2$ .

$θ = \frac{3 π}{8}$

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.

$2 θ - \frac{π}{2} = π + \frac{π}{4}$

Step 6

Solve for $θ$ .

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Step 6.1

Simplify $π + \frac{π}{4}$ .

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Step 6.1.1

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

$2 θ - \frac{π}{2} = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$2 θ - \frac{π}{2} = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 6.1.2.2

Combine the numerators over the common denominator.

$2 θ - \frac{π}{2} = \frac{π \cdot 4 + π}{4}$

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Move $4$ to the left of $π$ .

$2 θ - \frac{π}{2} = \frac{4 \cdot π + π}{4}$

Step 6.1.3.2

Add $4 π$ and $π$ .

$2 θ - \frac{π}{2} = \frac{5 π}{4}$

Step 6.2

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

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Step 6.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$2 θ = \frac{5 π}{4} + \frac{π}{2}$

Step 6.2.2

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

$2 θ = \frac{5 π}{4} + \frac{π}{2} \cdot \frac{2}{2}$

Step 6.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$2 θ = \frac{5 π}{4} + \frac{π \cdot 2}{2 \cdot 2}$

Step 6.2.3.2

Multiply $2$ by $2$ .

$2 θ = \frac{5 π}{4} + \frac{π \cdot 2}{4}$

Step 6.2.4

Combine the numerators over the common denominator.

$2 θ = \frac{5 π + π \cdot 2}{4}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Move $2$ to the left of $π$ .

$2 θ = \frac{5 π + 2 \cdot π}{4}$

Step 6.2.5.2

Add $5 π$ and $2 π$ .

$2 θ = \frac{7 π}{4}$

Step 6.3

Divide each term in $2 θ = \frac{7 π}{4}$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 θ = \frac{7 π}{4}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{7 π}{4}}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{7 π}{4}}{2}$

Step 6.3.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{7 π}{4}}{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{7 π}{4} \cdot \frac{1}{2}$

Step 6.3.3.2

Multiply $\frac{7 π}{4} \cdot \frac{1}{2}$ .

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Step 6.3.3.2.1

Multiply $\frac{7 π}{4}$ by $\frac{1}{2}$ .

$θ = \frac{7 π}{4 \cdot 2}$

Step 6.3.3.2.2

Multiply $4$ by $2$ .

$θ = \frac{7 π}{8}$

Step 7

Find the period of $\cot (2 θ - \frac{π}{2})$ .

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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 $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 8

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

$θ = \frac{3 π}{8} + \frac{π n}{2}, \frac{7 π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 9

Consolidate the answers.

$θ = \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$