Precalculus Examples

$\sec^{2} (x) - 8 \sec (x) = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \sec (x)$ . Substitute $u$ for all occurrences of $\sec (x)$ .

$u^{2} - 8 u = 0$

Step 1.2

Factor $u$ out of $u^{2} - 8 u$ .

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Step 1.2.1

Factor $u$ out of $u^{2}$ .

$u \cdot u - 8 u = 0$

Step 1.2.2

Factor $u$ out of $- 8 u$ .

$u \cdot u + u \cdot - 8 = 0$

Step 1.2.3

Factor $u$ out of $u \cdot u + u \cdot - 8$ .

$u (u - 8) = 0$

Step 1.3

Replace all occurrences of $u$ with $\sec (x)$ .

$\sec (x) (\sec (x) - 8) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sec (x) = 0$

$\sec (x) - 8 = 0$

Step 3

Set $\sec (x)$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\sec (x)$ equal to $0$ .

$\sec (x) = 0$

Step 3.2

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 4

Set $\sec (x) - 8$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\sec (x) - 8$ equal to $0$ .

$\sec (x) - 8 = 0$

Step 4.2

Solve $\sec (x) - 8 = 0$ for $x$ .

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Step 4.2.1

Add $8$ to both sides of the equation.

$\sec (x) = 8$

Step 4.2.2

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

$x = arcsec (8)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Evaluate $arcsec (8)$ .

$x = 1.44546849$

Step 4.2.4

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.

$x = 2 (3.14159265) - 1.44546849$

Step 4.2.5

Solve for $x$ .

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Step 4.2.5.1

Remove parentheses.

$x = 2 (3.14159265) - 1.44546849$

Step 4.2.5.2

Simplify $2 (3.14159265) - 1.44546849$ .

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Step 4.2.5.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.44546849$

Step 4.2.5.2.2

Subtract $1.44546849$ from $6.2831853$ .

$x = 4.83771681$

Step 4.2.6

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

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Step 4.2.6.1

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

$\frac{2 π}{| b |}$

Step 4.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 4.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.7

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

$x = 1.44546849 + 2 π n, 4.83771681 + 2 π n$ , for any integer $n$

Step 5

The final solution is all the values that make $\sec (x) (\sec (x) - 8) = 0$ true.

$x = 1.44546849 + 2 π n, 4.83771681 + 2 π n$ , for any integer $n$