Precalculus Examples

$\sec (90 - x) = 2$

Step 1

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

$90 - x = arcsec (2)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $arcsec (2)$ is $\frac{π}{3}$ .

$90 - x = \frac{π}{3}$

Step 3

Subtract $90$ from both sides of the equation.

$- x = \frac{π}{3} - 90$

Step 4

Divide each term in $- x = \frac{π}{3} - 90$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x = \frac{π}{3} - 90$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{π}{3}}{- 1} + \frac{- 90}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\frac{π}{3}}{- 1} + \frac{- 90}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{- 1} + \frac{- 90}{- 1}$

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 $\frac{\frac{π}{3}}{- 1}$ .

$x = - 1 \cdot \frac{π}{3} + \frac{- 90}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \frac{π}{3}$ as $- \frac{π}{3}$ .

$x = - \frac{π}{3} + \frac{- 90}{- 1}$

Step 4.3.1.3

Divide $- 90$ by $- 1$ .

$x = - \frac{π}{3} + 90$

Step 5

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$90 - x = 2 π - \frac{π}{3}$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 π - \frac{π}{3}$ .

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Step 6.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$90 - x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$90 - x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 6.1.2.2

Combine the numerators over the common denominator.

$90 - x = \frac{2 π \cdot 3 - π}{3}$

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Multiply $3$ by $2$ .

$90 - x = \frac{6 π - π}{3}$

Step 6.1.3.2

Subtract $π$ from $6 π$ .

$90 - x = \frac{5 π}{3}$

Step 6.2

Subtract $90$ from both sides of the equation.

$- x = \frac{5 π}{3} - 90$

Step 6.3

Divide each term in $- x = \frac{5 π}{3} - 90$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- x = \frac{5 π}{3} - 90$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{5 π}{3}}{- 1} + \frac{- 90}{- 1}$

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.

$\frac{x}{1} = \frac{\frac{5 π}{3}}{- 1} + \frac{- 90}{- 1}$

Step 6.3.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{3}}{- 1} + \frac{- 90}{- 1}$

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 $\frac{\frac{5 π}{3}}{- 1}$ .

$x = - 1 \cdot \frac{5 π}{3} + \frac{- 90}{- 1}$

Step 6.3.3.1.2

Rewrite $- 1 \cdot \frac{5 π}{3}$ as $- \frac{5 π}{3}$ .

$x = - \frac{5 π}{3} + \frac{- 90}{- 1}$

Step 6.3.3.1.3

Divide $- 90$ by $- 1$ .

$x = - \frac{5 π}{3} + 90$

Step 7

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

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Step 7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| - 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

Add $2 π$ to every negative angle to get positive angles.

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Step 8.1

Add $2 π$ to $- \frac{π}{3} + 90$ to find the positive angle.

$- \frac{π}{3} + 90 + 2 π$

Step 8.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 π \cdot \frac{3}{3} - \frac{π}{3} + 90$

Step 8.3

Combine fractions.

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Step 8.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3} + 90$

Step 8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3} + 90$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3} + 90$

Step 8.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3} + 90$

Step 8.5

Add $2 π$ to $- \frac{5 π}{3} + 90$ to find the positive angle.

$- \frac{5 π}{3} + 90 + 2 π$

Step 8.6

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 π \cdot \frac{3}{3} - \frac{5 π}{3} + 90$

Step 8.7

Combine fractions.

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Step 8.7.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{5 π}{3} + 90$

Step 8.7.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - 5 π}{3} + 90$

Step 8.8

Simplify the numerator.

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Step 8.8.1

Multiply $3$ by $2$ .

$\frac{6 π - 5 π}{3} + 90$

Step 8.8.2

Subtract $5 π$ from $6 π$ .

$\frac{π}{3} + 90$

Step 8.9

List the new angles.

$x = \frac{π}{3} + 90$

Step 9

The period of the $\sec (90 - x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{3} + 90 + 2 π n, \frac{5 π}{3} + 90 + 2 π n$ , for any integer $n$