Find All Complex Solutions 4sec(theta)+6=-2

$4 \sec (θ) + 6 = - 2$

Step 1

Move all terms not containing $\sec (θ)$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$4 \sec (θ) = - 2 - 6$

Subtract $6$ from $- 2$ .

$4 \sec (θ) = - 8$

Step 2

Divide each term in $4 \sec (θ) = - 8$ by $4$ and simplify.

Divide each term in $4 \sec (θ) = - 8$ by $4$ .

$\frac{4 \sec (θ)}{4} = \frac{- 8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 \sec (θ)}{4} = \frac{- 8}{4}$

Divide $\sec (θ)$ by $1$ .

$\sec (θ) = \frac{- 8}{4}$

Simplify the right side.

Divide $- 8$ by $4$ .

$\sec (θ) = - 2$

Step 3

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

$θ = arcsec (- 2)$

Step 4

Simplify the right side.

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

$θ = \frac{2 π}{3}$

Step 5

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.

$θ = 2 π - \frac{2 π}{3}$

Step 6

Simplify $2 π - \frac{2 π}{3}$ .

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

$θ = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Combine fractions.

Combine $2 π$ and $\frac{3}{3}$ .

$θ = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 3 - 2 π}{3}$

Simplify the numerator.

Multiply $3$ by $2$ .

$θ = \frac{6 π - 2 π}{3}$

Subtract $2 π$ from $6 π$ .

$θ = \frac{4 π}{3}$

Step 7

Find the period of $\sec (θ)$ .

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 8

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

$θ = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Find All Complex Solutions 4sec(theta)+6=-2

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