Trigonometry Examples

$\csc (x) - 1 = \frac{3}{2} \cdot \csc (x)$

Step 1

Simplify $\frac{3}{2} \csc (x)$ .

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Step 1.1

Rewrite.

$\csc (x) - 1 = 0 + 0 + \frac{3}{2} \csc (x)$

Step 1.2

Simplify by adding zeros.

$\csc (x) - 1 = \frac{3}{2} \csc (x)$

Step 1.3

Combine $\frac{3}{2}$ and $\csc (x)$ .

$\csc (x) - 1 = \frac{3 \csc (x)}{2}$

Step 2

Move all terms containing $\csc (x)$ to the left side of the equation.

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Step 2.1

Subtract $\frac{3 \csc (x)}{2}$ from both sides of the equation.

$\csc (x) - 1 - \frac{3 \csc (x)}{2} = 0$

Step 2.2

To write $\csc (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\csc (x) \cdot \frac{2}{2} - \frac{3 \csc (x)}{2} - 1 = 0$

Step 2.3

Combine $\csc (x)$ and $\frac{2}{2}$ .

$\frac{\csc (x) \cdot 2}{2} - \frac{3 \csc (x)}{2} - 1 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\csc (x) \cdot 2 - 3 \csc (x)}{2} - 1 = 0$

Step 2.5

Subtract $3 \csc (x)$ from $\csc (x) \cdot 2$ .

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Step 2.5.1

Reorder $\csc (x)$ and $2$ .

$\frac{2 \cdot \csc (x) - 3 \csc (x)}{2} - 1 = 0$

Step 2.5.2

Subtract $3 \csc (x)$ from $2 \cdot \csc (x)$ .

$\frac{- \csc (x)}{2} - 1 = 0$

Step 2.6

Move the negative in front of the fraction.

$- \frac{\csc (x)}{2} - 1 = 0$

Step 3

Add $1$ to both sides of the equation.

$- \frac{\csc (x)}{2} = 1$

Step 4

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{\csc (x)}{2}) = - 2 \cdot 1$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- 2 (- \frac{\csc (x)}{2})$ .

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Step 5.1.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{\csc (x)}{2}$ into the numerator.

$- 2 \frac{- \csc (x)}{2} = - 2 \cdot 1$

Step 5.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \csc (x)}{2} = - 2 \cdot 1$

Step 5.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \csc (x)}{2} = - 2 \cdot 1$

Step 5.1.1.1.4

Rewrite the expression.

$- - \csc (x) = - 2 \cdot 1$

Step 5.1.1.2

Multiply.

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Step 5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 \csc (x) = - 2 \cdot 1$

Step 5.1.1.2.2

Multiply $\csc (x)$ by $1$ .

$\csc (x) = - 2 \cdot 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 2$ by $1$ .

$\csc (x) = - 2$

Step 6

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

$x = arccsc (- 2)$

Step 7

Simplify the right side.

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Step 7.1

The exact value of $arccsc (- 2)$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 8

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{6} + π$

Step 9

Simplify the expression to find the second solution.

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Step 9.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 9.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 10.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 10.4

Divide $2 π$ by $1$ .

$2 π$

Step 11

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

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Step 11.1

Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 11.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 11.3

Combine fractions.

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Step 11.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 11.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 11.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 11.5

List the new angles.

$x = \frac{11 π}{6}$

Step 12

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$