Solve for x sec((2pi)/3+x)=2

$\sec (\frac{2 π}{3} + x) = 2$

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

$\frac{2 π}{3} + x = arcsec (2)$

Simplify the right side.

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

$\frac{2 π}{3} + x = \frac{π}{3}$

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

Subtract $\frac{2 π}{3}$ from both sides of the equation.

$x = \frac{π}{3} - \frac{2 π}{3}$

Combine the numerators over the common denominator.

$x = \frac{π - 2 π}{3}$

Subtract $2 π$ from $π$ .

$x = \frac{- π}{3}$

Move the negative in front of the fraction.

$x = - \frac{π}{3}$

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.

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

Solve for $x$ .

Simplify $2 π - \frac{π}{3}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{2 π}{3} + x = \frac{6 π - π}{3}$

Subtract $π$ from $6 π$ .

$\frac{2 π}{3} + x = \frac{5 π}{3}$

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

Subtract $\frac{2 π}{3}$ from both sides of the equation.

$x = \frac{5 π}{3} - \frac{2 π}{3}$

Combine the numerators over the common denominator.

$x = \frac{5 π - 2 π}{3}$

Subtract $2 π$ from $5 π$ .

$x = \frac{3 π}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x = \frac{3 π}{3}$

Divide $π$ by $1$ .

$x = π$

Find the period of $\sec (\frac{2 π}{3} + x)$ .

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.

$\frac{2 π}{| 1 |}$

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 $1$ .

$2 π$

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

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

$- \frac{π}{3} + 2 π$

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

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Combine fractions.

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

List the new angles.

$x = \frac{5 π}{3}$

The period of the $\sec (\frac{2 π}{3} + x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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

Solve for x sec((2pi)/3+x)=2

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