Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Subtract $2$ from both sides of the equation.

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

Step 4

Multiply both sides of the equation by $2$ .

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

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.

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

Step 5.1.1.2

Rewrite the expression.

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 (y - 2)$ .

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Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.1.2

Multiply $2$ by $- 2$ .

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

Step 6

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

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

Step 7

Multiply both sides of the equation by $2$ .

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

Step 8

Simplify the left side.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Cancel the common factor.

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

Step 8.1.2

Rewrite the expression.

$x = 2 \arcsin (2 y - 4)$