Solve for x sec(x+81)=2

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Trigonometry

$\sec (x + 81) = 2$

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

$x + 81 = arcsec (2)$

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

$x + 81 = \frac{π}{3}$

Subtract $81$ from both sides of the equation.

$x = \frac{π}{3} - 81$

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.

$x + 81 = 2 π - \frac{π}{3}$

Simplify the expression to find the second solution.

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Simplify $2 π - \frac{π}{3}$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x + 81 = \frac{2 π}{1} \cdot \frac{3}{3} - \frac{π}{3}$

Write each expression with a common denominator of $3$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x + 81 = \frac{2 π \cdot 3}{1 \cdot 3} - \frac{π}{3}$

Multiply $3$ by $1$ .

$x + 81 = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Combine the numerators over the common denominator.

$x + 81 = \frac{2 π \cdot 3 - π}{3}$

Simplify the numerator.

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Multiply $3$ by $2$ .

$x + 81 = \frac{6 π - π}{3}$

Subtract $π$ from $6 π$ .

$x + 81 = \frac{5 π}{3}$

Subtract $81$ from both sides of the equation.

$x = \frac{5 π}{3} - 81$

Add $2 π$ to $\frac{5 π}{3} - 81$ until the angle falls between $0$ and $2 π$ . In this case, $2 π$ needs to be added $13$ times.

$x = \frac{5 π}{3} - 81 + 13 (2 π)$

Simplify.

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Multiply $2$ by $13$ .

$x = \frac{5 π}{3} - 81 + 26 π$

To write $\frac{26 π}{1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{5 π}{3} + \frac{26 π}{1} \cdot \frac{3}{3} - 81$

Write each expression with a common denominator of $3$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{5 π}{3} + \frac{26 π \cdot 3}{1 \cdot 3} - 81$

Multiply $3$ by $1$ .

$x = \frac{5 π}{3} + \frac{26 π \cdot 3}{3} - 81$

Combine the numerators over the common denominator.

$x = \frac{5 π + 26 π \cdot 3}{3} - 81$

Simplify the numerator.

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Multiply $3$ by $26$ .

$x = \frac{5 π + 78 π}{3} - 81$

Add $5 π$ and $78 π$ .

$x = \frac{83 π}{3} - 81$

Find the period.

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

Solve the equation.

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\sec (x + 81)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{3} - 81 + 2 π n, \frac{83 π}{3} - 81 + 2 π n$ , for any integer $n$

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