Trigonometry Examples

$y = 3 \sin (2 x - \frac{π}{4}) + 2$

Step 1

Rewrite the equation as $3 \sin (2 x - \frac{π}{4}) + 2 = y$ .

$3 \sin (2 x - \frac{π}{4}) + 2 = y$

Step 2

Subtract $2$ from both sides of the equation.

$3 \sin (2 x - \frac{π}{4}) = y - 2$

Step 3

Divide each term in $3 \sin (2 x - \frac{π}{4}) = y - 2$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 \sin (2 x - \frac{π}{4}) = y - 2$ by $3$ .

$\frac{3 \sin (2 x - \frac{π}{4})}{3} = \frac{y}{3} + \frac{- 2}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 \sin (2 x - \frac{π}{4})}{3} = \frac{y}{3} + \frac{- 2}{3}$

Step 3.2.1.2

Divide $\sin (2 x - \frac{π}{4})$ by $1$ .

$\sin (2 x - \frac{π}{4}) = \frac{y}{3} + \frac{- 2}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$\sin (2 x - \frac{π}{4}) = \frac{y}{3} - \frac{2}{3}$

Step 4

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

$2 x - \frac{π}{4} = \arcsin (\frac{y}{3} - \frac{2}{3})$

Step 5

Add $\frac{π}{4}$ to both sides of the equation.

$2 x = \arcsin (\frac{y}{3} - \frac{2}{3}) + \frac{π}{4}$

Step 6

Divide each term in $2 x = \arcsin (\frac{y}{3} - \frac{2}{3}) + \frac{π}{4}$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = \arcsin (\frac{y}{3} - \frac{2}{3}) + \frac{π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (\frac{y}{3} - \frac{2}{3})}{2} + \frac{\frac{π}{4}}{2}$

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.

$\frac{2 x}{2} = \frac{\arcsin (\frac{y}{3} - \frac{2}{3})}{2} + \frac{\frac{π}{4}}{2}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{y}{3} - \frac{2}{3})}{2} + \frac{\frac{π}{4}}{2}$

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.

$x = \frac{\arcsin (\frac{y}{3} - \frac{2}{3})}{2} + \frac{π}{4} \cdot \frac{1}{2}$

Step 6.3.1.2

Multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 6.3.1.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

$x = \frac{\arcsin (\frac{y}{3} - \frac{2}{3})}{2} + \frac{π}{4 \cdot 2}$

Step 6.3.1.2.2

Multiply $4$ by $2$ .

$x = \frac{\arcsin (\frac{y}{3} - \frac{2}{3})}{2} + \frac{π}{8}$