Solve for x sec(four x) equals two

$\sec (4 x) = 2$

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

$4 x = arcsec (2)$

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

$4 x = \frac{π}{3}$

First, divide each term by $4$ and simplify.

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Divide each term in $4 x = \frac{π}{3}$ by $4$ .

$\frac{4 x}{4} = \frac{π}{3} \cdot \frac{1}{4}$

The common factor should be canceled of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{3} \cdot \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{π}{3} \cdot \frac{1}{4}$

First, multiply $\frac{π}{3} \cdot \frac{1}{4}$ .

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Multiply $\frac{π}{3}$ and $\frac{1}{4}$ .

$x = \frac{π}{3 \cdot 4}$

Multiply $3$ by $4$ .

$x = \frac{π}{12}$

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

$4 x = 2 π - \frac{π}{3}$

Clarify the expression To calculate the second answer.

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

$4 x = \frac{2 π}{1} \cdot \frac{3}{3} - \frac{π}{3}$

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

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Add.

$\frac{2 π \cdot 3}{1 \cdot 3} - \frac{π}{3}$

Multiply $3$ by $1$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

$4 x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Add numerators over the common denominator.

$4 x = \frac{2 π \cdot 3 - π}{3}$

Multiply $3$ by $2$ .

$4 x = \frac{6 π - π}{3}$

Subtract $π$ from $6 π$ .

$4 x = \frac{5 π}{3}$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = \frac{5 π}{3}$ by $4$ .

$\frac{4 x}{4} = \frac{5 π}{3} \cdot \frac{1}{4}$

The common factor $4$ should be canceled.

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Cancel the common factor.

$\frac{4 x}{4} = \frac{5 π}{3} \cdot \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{5 π}{3} \cdot \frac{1}{4}$

Multiply $\frac{5 π}{3} \cdot \frac{1}{4}$ .

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Multiply $\frac{5 π}{3}$ and $\frac{1}{4}$ .

$x = \frac{5 π}{3 \cdot 4}$

Multiply $3$ by $4$ .

$x = \frac{5 π}{12}$

Calculate the period.

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Evaluate the period of function using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Find solution to the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

The common factor $2$ should be canceled and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

The common factors should be canceled.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Reformulate the expression.

$\frac{π}{2}$

The period of the $\sec (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in two directions at once.

$x = \frac{π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}$ , for any integer $n$

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