Precalculus Examples

csc (x) = - 1

$\csc (x) = - 1$

Step 1

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

x

$x$ from inside the cosecant.

x = arccsc (- 1)

$x = arccsc (- 1)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

The exact value of

arccsc (- 1)

$arccsc (- 1)$ is

- \frac{π}{2}

$- \frac{π}{2}$ .

x = - \frac{π}{2}

$x = - \frac{π}{2}$

x = - \frac{π}{2}

$x = - \frac{π}{2}$

Step 3

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

2 π

$2 π$ , to find a reference angle. Next, add this reference angle to

π

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

x = 2 π + \frac{π}{2} + π

$x = 2 π + \frac{π}{2} + π$

Step 4

Simplify the expression to find the second solution.

Tap for more steps...

Step 4.1

Subtract

2 π

$2 π$ from

2 π + \frac{π}{2} + π

$2 π + \frac{π}{2} + π$ .

x = 2 π + \frac{π}{2} + π - 2 π

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 4.2

The resulting angle of

\frac{3 π}{2}

$\frac{3 π}{2}$ is positive, less than

2 π

$2 π$ , and coterminal with

2 π + \frac{π}{2} + π

$2 π + \frac{π}{2} + π$ .

x = \frac{3 π}{2}

$x = \frac{3 π}{2}$

x = \frac{3 π}{2}

$x = \frac{3 π}{2}$

Step 5

Find the period of

csc (x)

$\csc (x)$ .

Tap for more steps...

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

Add

2 π

$2 π$ to every negative angle to get positive angles.

Tap for more steps...

Step 6.1

Add

2 π

$2 π$ to

- \frac{π}{2}

$- \frac{π}{2}$ to find the positive angle.

- \frac{π}{2} + 2 π

$- \frac{π}{2} + 2 π$

Step 6.2

To write

2 π

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

\frac{2}{2}

$\frac{2}{2}$ .

2 π \cdot \frac{2}{2} - \frac{π}{2}

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.3

Combine fractions.

Tap for more steps...

Step 6.3.1

Combine

2 π

$2 π$ and

\frac{2}{2}

$\frac{2}{2}$ .

\frac{2 π \cdot 2}{2} - \frac{π}{2}

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.3.2

Combine the numerators over the common denominator.

\frac{2 π \cdot 2 - π}{2}

$\frac{2 π \cdot 2 - π}{2}$

\frac{2 π \cdot 2 - π}{2}

$\frac{2 π \cdot 2 - π}{2}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Multiply

2

$2$ by

2

$2$ .

\frac{4 π - π}{2}

$\frac{4 π - π}{2}$

Step 6.4.2

Subtract

π

$π$ from

4 π

$4 π$ .

\frac{3 π}{2}

$\frac{3 π}{2}$

\frac{3 π}{2}

$\frac{3 π}{2}$

Step 6.5

List the new angles.

x = \frac{3 π}{2}

$x = \frac{3 π}{2}$

x = \frac{3 π}{2}

$x = \frac{3 π}{2}$

Step 7

The period of the

csc (x)

$\csc (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer

n

$n$

Step 8

Consolidate the answers.

x = \frac{3 π}{2} + 2 π n

$x = \frac{3 π}{2} + 2 π n$ , for any integer

n

$n$