Trigonometry Examples

$30 = 11 (\frac{π}{3} \cdot (\cos (x) + 1)) + 36$

Step 1

Rewrite the equation as $11 (\frac{π}{3}) \cdot (\cos (x) + 1) + 36 = 30$ .

$11 (\frac{π}{3}) \cdot (\cos (x) + 1) + 36 = 30$

Step 2

Simplify each term.

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Step 2.1

Combine $11$ and $\frac{π}{3}$ .

$\frac{11 π}{3} \cdot (\cos (x) + 1) + 36 = 30$

Step 2.2

Apply the distributive property.

$\frac{11 π}{3} \cos (x) + \frac{11 π}{3} \cdot 1 + 36 = 30$

Step 2.3

Combine $\frac{11 π}{3}$ and $\cos (x)$ .

$\frac{11 π \cos (x)}{3} + \frac{11 π}{3} \cdot 1 + 36 = 30$

Step 2.4

Multiply $\frac{11 π}{3}$ by $1$ .

$\frac{11 π \cos (x)}{3} + \frac{11 π}{3} + 36 = 30$

Step 3

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 3.1

Subtract $\frac{11 π}{3}$ from both sides of the equation.

$\frac{11 π \cos (x)}{3} + 36 = 30 - \frac{11 π}{3}$

Step 3.2

Subtract $36$ from both sides of the equation.

$\frac{11 π \cos (x)}{3} = 30 - \frac{11 π}{3} - 36$

Step 3.3

Subtract $36$ from $30$ .

$\frac{11 π \cos (x)}{3} = - 6 - \frac{11 π}{3}$

Step 4

Multiply both sides of the equation by $\frac{3}{11 π}$ .

$\frac{3}{11 π} \cdot \frac{11 π \cos (x)}{3} = \frac{3}{11 π} (- 6 - \frac{11 π}{3})$

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

Simplify $\frac{3}{11 π} \cdot \frac{11 π \cos (x)}{3}$ .

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Step 5.1.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{3}{11 π} \cdot \frac{11 π \cos (x)}{3} = \frac{3}{11 π} (- 6 - \frac{11 π}{3})$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{11 π} (11 π \cos (x)) = \frac{3}{11 π} (- 6 - \frac{11 π}{3})$

Step 5.1.1.2

Cancel the common factor of $11 π$ .

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Step 5.1.1.2.1

Factor $11 π$ out of $11 π \cos (x)$ .

$\frac{1}{11 π} (11 π \cos (x)) = \frac{3}{11 π} (- 6 - \frac{11 π}{3})$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{11 π} (11 π \cos (x)) = \frac{3}{11 π} (- 6 - \frac{11 π}{3})$

Step 5.1.1.2.3

Rewrite the expression.

$\cos (x) = \frac{3}{11 π} (- 6 - \frac{11 π}{3})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{3}{11 π} (- 6 - \frac{11 π}{3})$ .

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Step 5.2.1.1

Apply the distributive property.

$\cos (x) = \frac{3}{11 π} \cdot - 6 + \frac{3}{11 π} (- \frac{11 π}{3})$

Step 5.2.1.2

Multiply $\frac{3}{11 π} \cdot - 6$ .

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Step 5.2.1.2.1

Combine $\frac{3}{11 π}$ and $- 6$ .

$\cos (x) = \frac{3 \cdot - 6}{11 π} + \frac{3}{11 π} (- \frac{11 π}{3})$

Step 5.2.1.2.2

Multiply $3$ by $- 6$ .

$\cos (x) = \frac{- 18}{11 π} + \frac{3}{11 π} (- \frac{11 π}{3})$

Step 5.2.1.3

Cancel the common factor of $3$ .

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Step 5.2.1.3.1

Move the leading negative in $- \frac{11 π}{3}$ into the numerator.

$\cos (x) = \frac{- 18}{11 π} + \frac{3}{11 π} \cdot \frac{- 11 π}{3}$

Step 5.2.1.3.2

Cancel the common factor.

$\cos (x) = \frac{- 18}{11 π} + \frac{3}{11 π} \cdot \frac{- 11 π}{3}$

Step 5.2.1.3.3

Rewrite the expression.

$\cos (x) = \frac{- 18}{11 π} + \frac{1}{11 π} (- 11 π)$

Step 5.2.1.4

Cancel the common factor of $11 π$ .

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Step 5.2.1.4.1

Factor $11 π$ out of $- 11 π$ .

$\cos (x) = \frac{- 18}{11 π} + \frac{1}{11 π} (11 π (- 1))$

Step 5.2.1.4.2

Cancel the common factor.

$\cos (x) = \frac{- 18}{11 π} + \frac{1}{11 π} (11 π \cdot - 1)$

Step 5.2.1.4.3

Rewrite the expression.

$\cos (x) = \frac{- 18}{11 π} - 1$

Step 5.2.1.5

Move the negative in front of the fraction.

$\cos (x) = - \frac{18}{11 π} - 1$

Step 6

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

$x = \arccos (- \frac{18}{11 π} - 1)$

Step 7

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \arccos (- \frac{18}{11 π} - 1)$

Step 8

Find the period of $\cos (x)$ .

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Step 8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \arccos (- \frac{18}{11 π} - 1) + 2 π n, 2 π - \arccos (- \frac{18}{11 π} - 1) + 2 π n$ , for any integer $n$