Trigonometry Examples

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

Step 1

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

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

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cos (3 x + 2 π)) = 2 y$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 (\frac{1}{2} \cos (3 x + 2 π))$ .

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Step 3.1.1

Combine $\frac{1}{2}$ and $\cos (3 x + 2 π)$ .

$2 \frac{\cos (3 x + 2 π)}{2} = 2 y$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Cancel the common factor.

$2 \frac{\cos (3 x + 2 π)}{2} = 2 y$

Step 3.1.2.2

Rewrite the expression.

$\cos (3 x + 2 π) = 2 y$

Step 4

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

$3 x + 2 π = \arccos (2 y)$

Step 5

Subtract $2 π$ from both sides of the equation.

$3 x = \arccos (2 y) - 2 π$

Step 6

Divide each term in $3 x = \arccos (2 y) - 2 π$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 x = \arccos (2 y) - 2 π$ by $3$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

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