Solve for x sec(4x)=2

$\sec (4 x) = 2$

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 $\frac{π}{3}$ .

$4 x = \frac{π}{3}$

Divide each term by $4$ and simplify.

Tap for more steps...

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

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

Cancel the common factor of $4$ .

Tap for more steps...

Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{π}{3} \cdot \frac{1}{4}$

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

Tap for more steps...

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 first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$4 x = 2 π - \frac{π}{3}$

Simplify the expression to find the second solution.

Tap for more steps...

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 $1$ .

Tap for more steps...

Combine.

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

Multiply $3$ by $1$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

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

Combine the 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.

Tap for more steps...

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

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

Cancel the common factor of $4$ .

Tap for more steps...

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 $\frac{5 π}{3} \cdot \frac{1}{4}$ .

Tap for more steps...

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

$x = \frac{5 π}{3 \cdot 4}$

Multiply $3$ by $4$ .

$x = \frac{5 π}{12}$

Find the period.

Tap for more steps...

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 4 |}$

Solve the equation.

Tap for more steps...

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

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

Tap for more steps...

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite 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 both directions.

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

Do you need help with solving Solve for x sec(4x)=2? We can help you. You can write to our math experts in our application. The best solution for you is above on this page.

Expand Using Sum/Difference Formulas (tan(40)+tan(20))/(1-tan(40)tan(20))

Solve for x 10^x=0.1

Simplify (-12x^5)^-2

Factor 2x^4-x^3-51x^2+25x+25

Simplify (9-x^2)(3-x)

Solve for x sec(4x)=2

100% Correct Answer from a Real Math Expert