Precalculus Examples

$2 - 3 \cos (x) = 5 + 3 \cos (x)$

Step 1

Move all terms containing $\cos (x)$ to the left side of the equation.

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Step 1.1

Subtract $3 \cos (x)$ from both sides of the equation.

$2 - 3 \cos (x) - 3 \cos (x) = 5$

Step 1.2

Subtract $3 \cos (x)$ from $- 3 \cos (x)$ .

$2 - 6 \cos (x) = 5$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the equation.

$- 6 \cos (x) = 5 - 2$

Step 2.2

Subtract $2$ from $5$ .

$- 6 \cos (x) = 3$

Step 3

Divide each term in $- 6 \cos (x) = 3$ by $- 6$ and simplify.

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Step 3.1

Divide each term in $- 6 \cos (x) = 3$ by $- 6$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $3$ and $- 6$ .

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Step 3.3.1.1

Factor $3$ out of $3$ .

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

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $3$ out of $- 6$ .

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

Step 3.3.1.2.2

Cancel the common factor.

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

Step 3.3.1.2.3

Rewrite the expression.

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

Step 3.3.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 6

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.

$x = 2 π - \frac{2 π}{3}$

Step 7

Simplify $2 π - \frac{2 π}{3}$ .

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Step 7.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 7.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

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.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$