Trigonometry Examples

$y = \frac{2}{3} \cdot \cos (5 x)$

Step 1

Rewrite the equation as $\frac{2}{3} \cdot \cos (5 x) = y$ .

$\frac{2}{3} \cdot \cos (5 x) = y$

Step 2

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} (\frac{2}{3} \cos (5 x)) = \frac{3}{2} y$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{3}{2} (\frac{2}{3} \cos (5 x))$ .

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Step 3.1.1.1

Combine $\frac{2}{3}$ and $\cos (5 x)$ .

$\frac{3}{2} \cdot \frac{2 \cos (5 x)}{3} = \frac{3}{2} y$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 \cos (5 x)}{3} = \frac{3}{2} y$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{1}{2} (2 \cos (5 x)) = \frac{3}{2} y$

Step 3.1.1.3

Cancel the common factor of $2$ .

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Step 3.1.1.3.1

Factor $2$ out of $2 \cos (5 x)$ .

$\frac{1}{2} (2 \cos (5 x)) = \frac{3}{2} y$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{2} (2 \cos (5 x)) = \frac{3}{2} y$

Step 3.1.1.3.3

Rewrite the expression.

$\cos (5 x) = \frac{3}{2} y$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{3}{2}$ and $y$ .

$\cos (5 x) = \frac{3 y}{2}$

Step 4

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

$5 x = \arccos (\frac{3 y}{2})$

Step 5

Divide each term in $5 x = \arccos (\frac{3 y}{2})$ by $5$ and simplify.

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Step 5.1

Divide each term in $5 x = \arccos (\frac{3 y}{2})$ by $5$ .

$\frac{5 x}{5} = \frac{\arccos (\frac{3 y}{2})}{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\arccos (\frac{3 y}{2})}{5}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arccos (\frac{3 y}{2})}{5}$