Trigonometry Examples

$\csc (4 x) - 2 = 0$

Step 1

Add $2$ to both sides of the equation.

$\csc (4 x) = 2$

Step 2

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

$4 x = arccsc (2)$

Step 3

Simplify the right side.

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Step 3.1

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

$4 x = \frac{π}{6}$

Step 4

Divide each term in $4 x = \frac{π}{6}$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 x = \frac{π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{4}$

Step 4.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{4}$ .

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Step 4.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{4}$ .

$x = \frac{π}{6 \cdot 4}$

Step 4.3.2.2

Multiply $6$ by $4$ .

$x = \frac{π}{24}$

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.

$4 x = π - \frac{π}{6}$

Step 6

Solve for $x$ .

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Step 6.1

Simplify.

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Step 6.1.1

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

$4 x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 6.1.2

Combine $π$ and $\frac{6}{6}$ .

$4 x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 6.1.3

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 6 - π}{6}$

Step 6.1.4

Subtract $π$ from $π \cdot 6$ .

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Step 6.1.4.1

Reorder $π$ and $6$ .

$4 x = \frac{6 \cdot π - π}{6}$

Step 6.1.4.2

Subtract $π$ from $6 \cdot π$ .

$4 x = \frac{5 \cdot π}{6}$

Step 6.2

Divide each term in $4 x = \frac{5 \cdot π}{6}$ by $4$ and simplify.

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Step 6.2.1

Divide each term in $4 x = \frac{5 \cdot π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{5 \cdot π}{6}}{4}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{5 \cdot π}{6}}{4}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{6}}{4}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 \cdot π}{6} \cdot \frac{1}{4}$

Step 6.2.3.2

Multiply $\frac{5 π}{6} \cdot \frac{1}{4}$ .

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Step 6.2.3.2.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{4}$ .

$x = \frac{5 π}{6 \cdot 4}$

Step 6.2.3.2.2

Multiply $6$ by $4$ .

$x = \frac{5 π}{24}$

Step 7

Find the period of $\csc (4 x)$ .

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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 $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 7.4

Cancel the common factor of $2$ and $4$ .

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Step 7.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 7.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 7.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 8

The period of the $\csc (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{24} + \frac{π n}{2}, \frac{5 π}{24} + \frac{π n}{2}$ , for any integer $n$