Precalculus Examples

$\csc (3 x) - 7 = 0$

Step 1

Add $7$ to both sides of the equation.

$\csc (3 x) = 7$

Step 2

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

$3 x = arccsc (7)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccsc (7)$ .

$3 x = 0.14334756$

Step 4

Divide each term in $3 x = 0.14334756$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = 0.14334756$ by $3$ .

$\frac{3 x}{3} = \frac{0.14334756}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0.14334756}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.14334756}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0.14334756$ by $3$ .

$x = 0.04778252$

Step 5

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.

$3 x = (3.14159265) - 0.14334756$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $0.14334756$ from $3.14159265$ .

$3 x = 2.99824508$

Step 6.2

Divide each term in $3 x = 2.99824508$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 x = 2.99824508$ by $3$ .

$\frac{3 x}{3} = \frac{2.99824508}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2.99824508}{3}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.99824508}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $2.99824508$ by $3$ .

$x = 0.99941502$

Step 7

Find the period of $\csc (3 x)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 8

The period of the $\csc (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = 0.04778252 + \frac{2 π n}{3}, 0.99941502 + \frac{2 π n}{3}$ , for any integer $n$