Precalculus Examples

sec (4 x) = 2

$\sec (4 x) = 2$

Step 1

Take the inverse secant of both sides of the equation to extract

x

$x$ from inside the secant.

4 x = arcsec (2)

$4 x = arcsec (2)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arcsec (2)

$arcsec (2)$ is

\frac{π}{3}

$\frac{π}{3}$ .

4 x = \frac{π}{3}

$4 x = \frac{π}{3}$

4 x = \frac{π}{3}

$4 x = \frac{π}{3}$

Step 3

Divide each term in

4 x = \frac{π}{3}

$4 x = \frac{π}{3}$ by

4

$4$ and simplify.

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Step 3.1

Divide each term in

4 x = \frac{π}{3}

$4 x = \frac{π}{3}$ by

4

$4$ .

\frac{4 x}{4} = \frac{\frac{π}{3}}{4}

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

4

$4$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{3}}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{4}$

Step 3.3.2

Multiply

$\frac{π}{3} \cdot \frac{1}{4}$ .

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Step 3.3.2.1

Multiply

$\frac{π}{3}$ by

$\frac{1}{4}$ .

$x = \frac{π}{3 \cdot 4}$

Step 3.3.2.2

Multiply

$3$ by

$4$ .

$x = \frac{π}{12}$

Step 4

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.

$4 x = 2 π - \frac{π}{3}$

Step 5

Solve for

$x$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

Step 5.1.2

Combine

$2 π$ and

$\frac{3}{3}$ .

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

Step 5.1.3

Combine the numerators over the common denominator.

$4 x = \frac{2 π \cdot 3 - π}{3}$

Step 5.1.4

Multiply

$3$ by

$2$ .

$4 x = \frac{6 π - π}{3}$

Step 5.1.5

Subtract

$π$ from

$6 π$ .

$4 x = \frac{5 π}{3}$

Step 5.2

Divide each term in

$4 x = \frac{5 π}{3}$ by

$4$ and simplify.

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Step 5.2.1

Divide each term in

$4 x = \frac{5 π}{3}$ by

$4$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$4$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{5 π}{3}}{4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{3} \cdot \frac{1}{4}$

Step 5.2.3.2

Multiply

$\frac{5 π}{3} \cdot \frac{1}{4}$ .

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Step 5.2.3.2.1

Multiply

$\frac{5 π}{3}$ by

$\frac{1}{4}$ .

$x = \frac{5 π}{3 \cdot 4}$

Step 5.2.3.2.2

Multiply

$3$ by

$4$ .

$x = \frac{5 π}{12}$

Step 6

Find the period of

$\sec (4 x)$ .

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Step 6.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2

Replace

$b$ with

$4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 6.3

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

$0$ and

$4$ is

$4$ .

$\frac{2 π}{4}$

Step 6.4

Cancel the common factor of

$2$ and

$4$ .

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Step 6.4.1

Factor

$2$ out of

$2 π$ .

$\frac{2 (π)}{4}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor

$2$ out of

$4$ .

$\frac{2 π}{2 \cdot 2}$

Step 6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 7

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

$n$