Precalculus Examples

csc (x) = \sqrt{2}

$\csc (x) = \sqrt{2}$

Step 1

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

x

$x$ from inside the cosecant.

x = arccsc (\sqrt{2})

$x = arccsc (\sqrt{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arccsc (\sqrt{2})

$arccsc (\sqrt{2})$ is

\frac{π}{4}

$\frac{π}{4}$ .

x = \frac{π}{4}

$x = \frac{π}{4}$

x = \frac{π}{4}

$x = \frac{π}{4}$

Step 3

The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from

π

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

x = π - \frac{π}{4}

$x = π - \frac{π}{4}$

Step 4

Simplify

π - \frac{π}{4}

$π - \frac{π}{4}$ .

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Step 4.1

To write

π

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

\frac{4}{4}

$\frac{4}{4}$ .

x = π \cdot \frac{4}{4} - \frac{π}{4}

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

Step 4.2

Combine fractions.

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Step 4.2.1

Combine

π

$π$ and

\frac{4}{4}

$\frac{4}{4}$ .

x = \frac{π \cdot 4}{4} - \frac{π}{4}

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

Step 4.2.2

Combine the numerators over the common denominator.

x = \frac{π \cdot 4 - π}{4}

$x = \frac{π \cdot 4 - π}{4}$

x = \frac{π \cdot 4 - π}{4}

$x = \frac{π \cdot 4 - π}{4}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Move

4

$4$ to the left of

π

$π$ .

x = \frac{4 \cdot π - π}{4}

$x = \frac{4 \cdot π - π}{4}$

Step 4.3.2

Subtract

π

$π$ from

4 π

$4 π$ .

x = \frac{3 π}{4}

$x = \frac{3 π}{4}$

x = \frac{3 π}{4}

$x = \frac{3 π}{4}$

x = \frac{3 π}{4}

$x = \frac{3 π}{4}$

Step 5

Find the period of

csc (x)

$\csc (x)$ .

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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

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{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n

$x = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n$ , for any integer

n

$n$