Trigonometry Examples

$y = \frac{3}{2} \cdot \sin (x - \frac{π}{3})$

Step 1

Rewrite the equation as $\frac{3}{2} \cdot \sin (x - \frac{π}{3}) = y$ .

$\frac{3}{2} \cdot \sin (x - \frac{π}{3}) = y$

Step 2

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

$\frac{2}{3} (\frac{3}{2} \sin (x - \frac{π}{3})) = \frac{2}{3} y$

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 $\frac{2}{3} (\frac{3}{2} \sin (x - \frac{π}{3}))$ .

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Step 3.1.1.1

Combine $\frac{3}{2}$ and $\sin (x - \frac{π}{3})$ .

$\frac{2}{3} \cdot \frac{3 \sin (x - \frac{π}{3})}{2} = \frac{2}{3} y$

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.

$\frac{2}{3} \cdot \frac{3 \sin (x - \frac{π}{3})}{2} = \frac{2}{3} y$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{1}{3} (3 \sin (x - \frac{π}{3})) = \frac{2}{3} y$

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 $3 \sin (x - \frac{π}{3})$ .

$\frac{1}{3} (3 \sin (x - \frac{π}{3})) = \frac{2}{3} y$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{3} (3 \sin (x - \frac{π}{3})) = \frac{2}{3} y$

Step 3.1.1.3.3

Rewrite the expression.

$\sin (x - \frac{π}{3}) = \frac{2}{3} y$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

$\sin (x - \frac{π}{3}) = \frac{2 y}{3}$

Step 4

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

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

Step 5

Add $\frac{π}{3}$ to both sides of the equation.

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