Precalculus Examples

$\sec (x) \cot (x) = 4$ , $\csc (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\sec (x) \cot (x)$ .

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Step 1.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1.1

Reorder $\sec (x)$ and $\cot (x)$ .

$\cot (x) \sec (x) = 4$

Step 1.1.1.2

Rewrite $\sec (x) \cot (x)$ in terms of sines and cosines.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)} = 4$

Step 1.1.1.3

Cancel the common factors.

$\frac{1}{\sin (x)} = 4$

Step 1.1.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) = 4$

Step 2

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

$x = arccsc (4)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccsc (4)$ .

$x = 0.25268025$

Step 4

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

$x = (3.14159265) - 0.25268025$

Step 5

Solve for $x$ .

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Step 5.1

Remove parentheses.

$x = 3.14159265 - 0.25268025$

Step 5.2

Remove parentheses.

$x = (3.14159265) - 0.25268025$

Step 5.3

Subtract $0.25268025$ from $3.14159265$ .

$x = 2.88891239$

Step 6

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 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 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$x = 0.25268025 + 2 π n, 2.88891239 + 2 π n$ , for any integer $n$

Step 8

Take the base solution.

$x = 0.25268025$

Step 9

Replace the variable $x$ with $0.25268025$ in the expression.

$\csc (0.25268025)$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\csc (0.25268025)$

Decimal Form:

$4.000000000000 \dots$