Trigonometry Examples

$y = 2 \csc (\frac{π x}{2})$

Step 1

Rewrite the equation as $2 \csc (\frac{π x}{2}) = y$ .

$2 \csc (\frac{π x}{2}) = y$

Step 2

Divide each term in $2 \csc (\frac{π x}{2}) = y$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \csc (\frac{π x}{2}) = y$ by $2$ .

$\frac{2 \csc (\frac{π x}{2})}{2} = \frac{y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \csc (\frac{π x}{2})}{2} = \frac{y}{2}$

Step 2.2.1.2

Divide $\csc (\frac{π x}{2})$ by $1$ .

$\csc (\frac{π x}{2}) = \frac{y}{2}$

Step 3

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

$\frac{π x}{2} = arccsc (\frac{y}{2})$

Step 4

Multiply both sides of the equation by $\frac{2}{π}$ .

$\frac{2}{π} \cdot \frac{π x}{2} = \frac{2}{π} arccsc (\frac{y}{2})$

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 $\frac{2}{π} \cdot \frac{π x}{2}$ .

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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

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π x}{2} = \frac{2}{π} arccsc (\frac{y}{2})$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{2}{π} arccsc (\frac{y}{2})$

Step 5.1.1.2

Cancel the common factor of $π$ .

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Step 5.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{2}{π} arccsc (\frac{y}{2})$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{2}{π} arccsc (\frac{y}{2})$

Step 5.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} arccsc (\frac{y}{2})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Combine $\frac{2}{π}$ and $arccsc (\frac{y}{2})$ .

$x = \frac{2 arccsc (\frac{y}{2})}{π}$