Trigonometry Examples

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

Step 1

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

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

Step 2

Combine $\frac{1}{2}$ and $\sin (π (x - 1))$ .

$\frac{\sin (π (x - 1))}{2} + 3 = y$

Step 3

Subtract $3$ from both sides of the equation.

$\frac{\sin (π (x - 1))}{2} = y - 3$

Step 4

Multiply both sides of the equation by $2$ .

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

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

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

Step 5.1.1.2

Rewrite the expression.

$\sin (π (x - 1)) = 2 (y - 3)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 (y - 3)$ .

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Step 5.2.1.1

Apply the distributive property.

$\sin (π (x - 1)) = 2 y + 2 \cdot - 3$

Step 5.2.1.2

Multiply $2$ by $- 3$ .

$\sin (π (x - 1)) = 2 y - 6$

Step 6

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

$π (x - 1) = \arcsin (2 y - 6)$

Step 7

Simplify the left side.

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Step 7.1

Simplify $π (x - 1)$ .

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Step 7.1.1

Simplify by multiplying through.

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Step 7.1.1.1

Apply the distributive property.

$π x + π \cdot - 1 = \arcsin (2 y - 6)$

Step 7.1.1.2

Move $- 1$ to the left of $π$ .

$π x - 1 \cdot π = \arcsin (2 y - 6)$

Step 7.1.2

Rewrite $- 1 π$ as $- π$ .

$π x - π = \arcsin (2 y - 6)$

Step 8

Add $π$ to both sides of the equation.

$π x = \arcsin (2 y - 6) + π$

Step 9

Divide each term in $π x = \arcsin (2 y - 6) + π$ by $π$ and simplify.

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Step 9.1

Divide each term in $π x = \arcsin (2 y - 6) + π$ by $π$ .

$\frac{π x}{π} = \frac{\arcsin (2 y - 6)}{π} + \frac{π}{π}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $π$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{\arcsin (2 y - 6)}{π} + \frac{π}{π}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (2 y - 6)}{π} + \frac{π}{π}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Cancel the common factor of $π$ .

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Step 9.3.1.1

Cancel the common factor.

$x = \frac{\arcsin (2 y - 6)}{π} + \frac{π}{π}$

Step 9.3.1.2

Rewrite the expression.

$x = \frac{\arcsin (2 y - 6)}{π} + 1$