Solve for x 3sec(x)=-2 square root of 3

$3 \sec (x) = - 2 \sqrt{3}$

Step 1

Divide each term in $3 \sec (x) = - 2 \sqrt{3}$ by $3$ and simplify.

Divide each term in $3 \sec (x) = - 2 \sqrt{3}$ by $3$ .

$\frac{3 \sec (x)}{3} = \frac{- 2 \sqrt{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 \sec (x)}{3} = \frac{- 2 \sqrt{3}}{3}$

Divide $\sec (x)$ by $1$ .

$\sec (x) = \frac{- 2 \sqrt{3}}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$\sec (x) = - \frac{2 \sqrt{3}}{3}$

Step 2

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

$x = arcsec (- \frac{2 \sqrt{3}}{3})$

Step 3

Simplify the right side.

The exact value of $arcsec (- \frac{2 \sqrt{3}}{3})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

Step 4

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

$x = 2 π - \frac{5 π}{6}$

Step 5

Simplify $2 π - \frac{5 π}{6}$ .

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Combine fractions.

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Simplify the numerator.

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 6

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

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 $1$ .

$2 π$

Step 7

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

$x = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Solve for x 3sec(x)=-2 square root of 3

Related Questions

Support LCMGCF.com by disabling your adblocker.