Precalculus Examples

sec (x) = - 1

$\sec (x) = - 1$

Step 1

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

x

$x$ from inside the secant.

x = arcsec (- 1)

$x = arcsec (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arcsec (- 1)

$arcsec (- 1)$ is

π

$π$ .

x = π

$x = π$

x = π

$x = π$

Step 3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from

2 π

$2 π$ to find the solution in the third quadrant.

x = 2 π - π

$x = 2 π - π$

Step 4

Subtract

π

$π$ from

2 π

$2 π$ .

x = π

$x = π$

Step 5

Find the period of

sec (x)

$\sec (x)$ .

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Step 5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 6

The period of the

sec (x)

$\sec (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = π + 2 π n

$x = π + 2 π n$ , for any integer

n

$n$