Precalculus Examples

$- 2 = 9 (\cos (x - 4)) + 7$

Step 1

Rewrite the equation as $9 \cos (x - 4) + 7 = - 2$ .

$9 \cos (x - 4) + 7 = - 2$

Step 2

Move all terms not containing $\cos (x - 4)$ to the right side of the equation.

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Step 2.1

Subtract $7$ from both sides of the equation.

$9 \cos (x - 4) = - 2 - 7$

Step 2.2

Subtract $7$ from $- 2$ .

$9 \cos (x - 4) = - 9$

Step 3

Divide each term in $9 \cos (x - 4) = - 9$ by $9$ and simplify.

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Step 3.1

Divide each term in $9 \cos (x - 4) = - 9$ by $9$ .

$\frac{9 \cos (x - 4)}{9} = \frac{- 9}{9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{9 \cos (x - 4)}{9} = \frac{- 9}{9}$

Step 3.2.1.2

Divide $\cos (x - 4)$ by $1$ .

$\cos (x - 4) = \frac{- 9}{9}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 9$ by $9$ .

$\cos (x - 4) = - 1$

Step 4

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

$x - 4 = \arccos (- 1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (- 1)$ is $π$ .

$x - 4 = π$

Step 6

Add $4$ to both sides of the equation.

$x = π + 4$

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.

$x - 4 = 2 π - π$

Step 8

Solve for $x$ .

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Step 8.1

Subtract $π$ from $2 π$ .

$x - 4 = π$

Step 8.2

Add $4$ to both sides of the equation.

$x = π + 4$

Step 9

Find the period of $\cos (x - 4)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

The period of the $\cos (x - 4)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = π + 4 + 2 π n$ , for any integer $n$