Trigonometry Examples

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

Step 1

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

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

Step 2

Subtract $2$ from both sides of the equation.

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply both sides of the equation by $3$ .

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

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $3$ .

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Step 6.1.1.1

Cancel the common factor.

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

Step 6.1.1.2

Rewrite the expression.

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

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $3 (\arcsin (y - 2) + \frac{π}{3})$ .

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Step 6.2.1.1

Apply the distributive property.

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

Step 6.2.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.2.1

Cancel the common factor.

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

Step 6.2.1.2.2

Rewrite the expression.

$x = 3 \arcsin (y - 2) + π$

Step 7

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

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

Step 8

Multiply both sides of the equation by $3$ .

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

Step 9

Simplify both sides of the equation.

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Step 9.1

Simplify the left side.

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Step 9.1.1

Cancel the common factor of $3$ .

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Step 9.1.1.1

Cancel the common factor.

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

Step 9.1.1.2

Rewrite the expression.

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

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $3 (\arcsin (y - 2) + \frac{π}{3})$ .

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Step 9.2.1.1

To write $\arcsin (y - 2)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 9.2.1.2

Simplify terms.

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Step 9.2.1.2.1

Combine $\arcsin (y - 2)$ and $\frac{3}{3}$ .

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

Step 9.2.1.2.2

Combine the numerators over the common denominator.

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

Step 9.2.1.2.3

Cancel the common factor of $3$ .

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Step 9.2.1.2.3.1

Cancel the common factor.

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

Step 9.2.1.2.3.2

Rewrite the expression.

$x = \arcsin (y - 2) \cdot 3 + π$

Step 9.2.1.3

Move $3$ to the left of $\arcsin (y - 2)$ .

$x = 3 \arcsin (y - 2) + π$