Algebra Examples

$y = 2 \cos (b (x - 2)) - 1$

Step 1

Rewrite the equation as $2 \cos (b (x - 2)) - 1 = y$ .

$2 \cos (b (x - 2)) - 1 = y$

Step 2

Add $1$ to both sides of the equation.

$2 \cos (b (x - 2)) = y + 1$

Step 3

Divide each term in $2 \cos (b (x - 2)) = y + 1$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \cos (b (x - 2)) = y + 1$ by $2$ .

$\frac{2 \cos (b (x - 2))}{2} = \frac{y}{2} + \frac{1}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 \cos (b (x - 2))}{2} = \frac{y}{2} + \frac{1}{2}$

Step 3.2.1.2

Divide $\cos (b (x - 2))$ by $1$ .

$\cos (b (x - 2)) = \frac{y}{2} + \frac{1}{2}$

Step 4

Take the inverse cosine of both sides of the equation to extract $b$ from inside the cosine.

$b (x - 2) = \arccos (\frac{y}{2} + \frac{1}{2})$

Step 5

Simplify the left side.

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Step 5.1

Simplify $b (x - 2)$ .

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Step 5.1.1

Apply the distributive property.

$b x + b \cdot - 2 = \arccos (\frac{y}{2} + \frac{1}{2})$

Step 5.1.2

Move $- 2$ to the left of $b$ .

$b x - 2 b = \arccos (\frac{y}{2} + \frac{1}{2})$

Step 6

Factor $b$ out of $b x - 2 b$ .

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Step 6.1

Factor $b$ out of $b x$ .

$b (x) - 2 b = \arccos (\frac{y}{2} + \frac{1}{2})$

Step 6.2

Factor $b$ out of $- 2 b$ .

$b (x) + b \cdot - 2 = \arccos (\frac{y}{2} + \frac{1}{2})$

Step 6.3

Factor $b$ out of $b (x) + b \cdot - 2$ .

$b (x - 2) = \arccos (\frac{y}{2} + \frac{1}{2})$

Step 7

Divide each term in $b (x - 2) = \arccos (\frac{y}{2} + \frac{1}{2})$ by $x - 2$ and simplify.

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Step 7.1

Divide each term in $b (x - 2) = \arccos (\frac{y}{2} + \frac{1}{2})$ by $x - 2$ .

$\frac{b (x - 2)}{x - 2} = \frac{\arccos (\frac{y}{2} + \frac{1}{2})}{x - 2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $x - 2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{b (x - 2)}{x - 2} = \frac{\arccos (\frac{y}{2} + \frac{1}{2})}{x - 2}$

Step 7.2.1.2

Divide $b$ by $1$ .

$b = \frac{\arccos (\frac{y}{2} + \frac{1}{2})}{x - 2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Combine the numerators over the common denominator.

$b = \frac{\arccos (\frac{y + 1}{2})}{x - 2}$