Trigonometry Examples

$y = 2 \sin (2 x - \frac{π}{2})$

Step 1

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

$2 \sin (2 x - \frac{π}{2}) = y$

Step 2

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

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Step 2.1

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

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

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.

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

Step 2.2.1.2

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

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

Step 3

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

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

Step 4

Add $\frac{π}{2}$ to both sides of the equation.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{\arcsin (\frac{y}{2})}{2} + \frac{π}{2} \cdot \frac{1}{2}$

Step 5.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 5.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

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

Step 5.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{\arcsin (\frac{y}{2})}{2} + \frac{π}{4}$