Trigonometry Examples

$r = \frac{2}{1 + \cos (x)}$

Step 1

Rewrite the equation as $\frac{2}{1 + \cos (x)} = r$ .

$\frac{2}{1 + \cos (x)} = r$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1 + \cos (x), 1$

Step 2.2

Remove parentheses.

$1 + \cos (x), 1$

Step 2.3

The LCM of one and any expression is the expression.

$1 + \cos (x)$

Step 3

Multiply each term in $\frac{2}{1 + \cos (x)} = r$ by $1 + \cos (x)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{2}{1 + \cos (x)} = r$ by $1 + \cos (x)$ .

$\frac{2}{1 + \cos (x)} (1 + \cos (x)) = r (1 + \cos (x))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $1 + \cos (x)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2}{1 + \cos (x)} (1 + \cos (x)) = r (1 + \cos (x))$

Step 3.2.1.2

Rewrite the expression.

$2 = r (1 + \cos (x))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$2 = r \cdot 1 + r \cos (x)$

Step 3.3.2

Multiply $r$ by $1$ .

$2 = r + r \cos (x)$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $r + r \cos (x) = 2$ .

$r + r \cos (x) = 2$

Step 4.2

Subtract $r$ from both sides of the equation.

$r \cos (x) = 2 - r$

Step 4.3

Divide each term in $r \cos (x) = 2 - r$ by $r$ and simplify.

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Step 4.3.1

Divide each term in $r \cos (x) = 2 - r$ by $r$ .

$\frac{r \cos (x)}{r} = \frac{2}{r} + \frac{- r}{r}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $r$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{r \cos (x)}{r} = \frac{2}{r} + \frac{- r}{r}$

Step 4.3.2.1.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{2}{r} + \frac{- r}{r}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $r$ .

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Step 4.3.3.1.1

Cancel the common factor.

$\cos (x) = \frac{2}{r} + \frac{- r}{r}$

Step 4.3.3.1.2

Divide $- 1$ by $1$ .

$\cos (x) = \frac{2}{r} - 1$

Step 5

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

$x = \arccos (\frac{2}{r} - 1)$