Trigonometry Examples

$y = \frac{5}{2} \cdot \sin (2 x + \frac{π}{2})$

Step 1

Rewrite the equation as $\frac{5}{2} \cdot \sin (2 x + \frac{π}{2}) = y$ .

$\frac{5}{2} \cdot \sin (2 x + \frac{π}{2}) = y$

Step 2

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} (\frac{5}{2} \sin (2 x + \frac{π}{2})) = \frac{2}{5} y$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2}{5} (\frac{5}{2} \sin (2 x + \frac{π}{2}))$ .

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Step 3.1.1.1

Combine $\frac{5}{2}$ and $\sin (2 x + \frac{π}{2})$ .

$\frac{2}{5} \cdot \frac{5 \sin (2 x + \frac{π}{2})}{2} = \frac{2}{5} y$

Step 3.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{2}{5} \cdot \frac{5 \sin (2 x + \frac{π}{2})}{2} = \frac{2}{5} y$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{1}{5} (5 \sin (2 x + \frac{π}{2})) = \frac{2}{5} y$

Step 3.1.1.3

Cancel the common factor of $5$ .

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Step 3.1.1.3.1

Factor $5$ out of $5 \sin (2 x + \frac{π}{2})$ .

$\frac{1}{5} (5 \sin (2 x + \frac{π}{2})) = \frac{2}{5} y$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{5} (5 \sin (2 x + \frac{π}{2})) = \frac{2}{5} y$

Step 3.1.1.3.3

Rewrite the expression.

$\sin (2 x + \frac{π}{2}) = \frac{2}{5} y$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{2}{5}$ and $y$ .

$\sin (2 x + \frac{π}{2}) = \frac{2 y}{5}$

Step 4

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

$2 x + \frac{π}{2} = \arcsin (\frac{2 y}{5})$

Step 5

Subtract $\frac{π}{2}$ from both sides of the equation.

$2 x = \arcsin (\frac{2 y}{5}) - \frac{π}{2}$

Step 6

Divide each term in $2 x = \arcsin (\frac{2 y}{5}) - \frac{π}{2}$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = \arcsin (\frac{2 y}{5}) - \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (\frac{2 y}{5})}{2} + \frac{- \frac{π}{2}}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\arcsin (\frac{2 y}{5})}{2} + \frac{- \frac{π}{2}}{2}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{2 y}{5})}{2} + \frac{- \frac{π}{2}}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{\arcsin (\frac{2 y}{5})}{2} - \frac{π}{2} \cdot \frac{1}{2}$

Step 6.3.1.2

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

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Step 6.3.1.2.1

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

$x = \frac{\arcsin (\frac{2 y}{5})}{2} - \frac{π}{2 \cdot 2}$

Step 6.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{\arcsin (\frac{2 y}{5})}{2} - \frac{π}{4}$