Trigonometry Examples

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

Step 1

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

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

Step 2

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

$\frac{\cos (3 x - π)}{3} + 2 = y$

Step 3

Subtract $2$ from both sides of the equation.

$\frac{\cos (3 x - π)}{3} = y - 2$

Step 4

Multiply both sides of the equation by $3$ .

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

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 $3$ .

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Step 5.1.1.1

Cancel the common factor.

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

Step 5.1.1.2

Rewrite the expression.

$\cos (3 x - π) = 3 (y - 2)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $3 (y - 2)$ .

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Step 5.2.1.1

Apply the distributive property.

$\cos (3 x - π) = 3 y + 3 \cdot - 2$

Step 5.2.1.2

Multiply $3$ by $- 2$ .

$\cos (3 x - π) = 3 y - 6$

Step 6

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

$3 x - π = \arccos (3 y - 6)$

Step 7

Add $π$ to both sides of the equation.

$3 x = \arccos (3 y - 6) + π$

Step 8

Divide each term in $3 x = \arccos (3 y - 6) + π$ by $3$ and simplify.

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Step 8.1

Divide each term in $3 x = \arccos (3 y - 6) + π$ by $3$ .

$\frac{3 x}{3} = \frac{\arccos (3 y - 6)}{3} + \frac{π}{3}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\arccos (3 y - 6)}{3} + \frac{π}{3}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arccos (3 y - 6)}{3} + \frac{π}{3}$