Precalculus Examples

$\cot (90 - x) = 1$

Step 1

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

$90 - x = arccot (1)$

Step 2

Simplify the right side.

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Step 2.1

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

$90 - x = \frac{π}{4}$

Step 3

Subtract $90$ from both sides of the equation.

$- x = \frac{π}{4} - 90$

Step 4

Divide each term in $- x = \frac{π}{4} - 90$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x = \frac{π}{4} - 90$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{π}{4}}{- 1} + \frac{- 90}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\frac{π}{4}}{- 1} + \frac{- 90}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{- 1} + \frac{- 90}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\frac{π}{4}}{- 1}$ .

$x = - 1 \cdot \frac{π}{4} + \frac{- 90}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \frac{π}{4}$ as $- \frac{π}{4}$ .

$x = - \frac{π}{4} + \frac{- 90}{- 1}$

Step 4.3.1.3

Divide $- 90$ by $- 1$ .

$x = - \frac{π}{4} + 90$

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.

$90 - x = π + \frac{π}{4}$

Step 6

Solve for $x$ .

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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 $\frac{4}{4}$ .

$90 - x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$90 - x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 6.1.2.2

Combine the numerators over the common denominator.

$90 - x = \frac{π \cdot 4 + π}{4}$

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Move $4$ to the left of $π$ .

$90 - x = \frac{4 \cdot π + π}{4}$

Step 6.1.3.2

Add $4 π$ and $π$ .

$90 - x = \frac{5 π}{4}$

Step 6.2

Subtract $90$ from both sides of the equation.

$- x = \frac{5 π}{4} - 90$

Step 6.3

Divide each term in $- x = \frac{5 π}{4} - 90$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- x = \frac{5 π}{4} - 90$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{5 π}{4}}{- 1} + \frac{- 90}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\frac{5 π}{4}}{- 1} + \frac{- 90}{- 1}$

Step 6.3.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{4}}{- 1} + \frac{- 90}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

Move the negative one from the denominator of $\frac{\frac{5 π}{4}}{- 1}$ .

$x = - 1 \cdot \frac{5 π}{4} + \frac{- 90}{- 1}$

Step 6.3.3.1.2

Rewrite $- 1 \cdot \frac{5 π}{4}$ as $- \frac{5 π}{4}$ .

$x = - \frac{5 π}{4} + \frac{- 90}{- 1}$

Step 6.3.3.1.3

Divide $- 90$ by $- 1$ .

$x = - \frac{5 π}{4} + 90$

Step 7

Find the period of $\cot (90 - x)$ .

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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 $- 1$ in the formula for period.

$\frac{π}{| - 1 |}$

Step 7.3

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

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

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

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Step 8.1

Add $π$ to $- \frac{π}{4} + 90$ to find the positive angle.

$- \frac{π}{4} + 90 + π$

Step 8.2

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

$π \cdot \frac{4}{4} - \frac{π}{4} + 90$

Step 8.3

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4} + 90$

Step 8.4

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4} + 90$

Step 8.5

Subtract $π$ from $π \cdot 4$ .

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Step 8.5.1

Reorder $π$ and $4$ .

$\frac{4 \cdot π - π}{4} + 90$

Step 8.5.2

Subtract $π$ from $4 \cdot π$ .

$\frac{3 \cdot π}{4} + 90$

$\frac{3 π}{4} + 90$

Step 8.6

Add $π$ to $- \frac{5 π}{4} + 90$ to find the positive angle.

$- \frac{5 π}{4} + 90 + π$

Step 8.7

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

$π \cdot \frac{4}{4} - \frac{5 π}{4} + 90$

Step 8.8

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{5 π}{4} + 90$

Step 8.9

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - 5 π}{4} + 90$

Step 8.10

Subtract $5 π$ from $π \cdot 4$ .

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Step 8.10.1

Reorder $π$ and $4$ .

$\frac{4 \cdot π - 5 π}{4} + 90$

Step 8.10.2

Subtract $5 π$ from $4 \cdot π$ .

$\frac{- π}{4} + 90$

Step 8.11

Move the negative in front of the fraction.

$- \frac{π}{4} + 90$

Step 8.12

List the new angles.

$x = - \frac{π}{4} + 90$

Step 9

The period of the $\cot (90 - x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = - \frac{π}{4} + 90 + π n, \frac{3 π}{4} + 90 + π n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = - \frac{π}{4} + 90 + π n$ , for any integer $n$