Precalculus Examples

$y = \frac{3}{2} \cdot \sin (5 x + 1)$

Step 1

Rewrite the equation as $\frac{3}{2} \cdot \sin (5 x + 1) = y$ .

$\frac{3}{2} \cdot \sin (5 x + 1) = y$

Step 2

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} (\frac{3}{2} \sin (5 x + 1)) = \frac{2}{3} y$

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 $\frac{2}{3} (\frac{3}{2} \sin (5 x + 1))$ .

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Step 3.1.1.1

Combine $\frac{3}{2}$ and $\sin (5 x + 1)$ .

$\frac{2}{3} \cdot \frac{3 \sin (5 x + 1)}{2} = \frac{2}{3} y$

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.

$\frac{2}{3} \cdot \frac{3 \sin (5 x + 1)}{2} = \frac{2}{3} y$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{1}{3} (3 \sin (5 x + 1)) = \frac{2}{3} y$

Step 3.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.3.1

Factor $3$ out of $3 \sin (5 x + 1)$ .

$\frac{1}{3} (3 \sin (5 x + 1)) = \frac{2}{3} y$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{3} (3 \sin (5 x + 1)) = \frac{2}{3} y$

Step 3.1.1.3.3

Rewrite the expression.

$\sin (5 x + 1) = \frac{2}{3} y$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{2}{3}$ and $y$ .

$\sin (5 x + 1) = \frac{2 y}{3}$

Step 4

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

$5 x + 1 = \arcsin (\frac{2 y}{3})$

Step 5

Subtract $1$ from both sides of the equation.

$5 x = \arcsin (\frac{2 y}{3}) - 1$

Step 6

Divide each term in $5 x = \arcsin (\frac{2 y}{3}) - 1$ by $5$ and simplify.

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Step 6.1

Divide each term in $5 x = \arcsin (\frac{2 y}{3}) - 1$ by $5$ .

$\frac{5 x}{5} = \frac{\arcsin (\frac{2 y}{3})}{5} + \frac{- 1}{5}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\arcsin (\frac{2 y}{3})}{5} + \frac{- 1}{5}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{2 y}{3})}{5} + \frac{- 1}{5}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = \frac{\arcsin (\frac{2 y}{3})}{5} - \frac{1}{5}$