Algebra Examples

$3 \cos (x + \frac{π}{6})$

Step 1

Set $3 \cos (x + \frac{π}{6})$ equal to $0$ .

$3 \cos (x + \frac{π}{6}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $3 \cos (x + \frac{π}{6}) = 0$ by $3$ and simplify.

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Step 2.1.1

Divide each term in $3 \cos (x + \frac{π}{6}) = 0$ by $3$ .

$\frac{3 \cos (x + \frac{π}{6})}{3} = \frac{0}{3}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $3$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{3 \cos (x + \frac{π}{6})}{3} = \frac{0}{3}$

Step 2.1.2.1.2

Divide $\cos (x + \frac{π}{6})$ by $1$ .

$\cos (x + \frac{π}{6}) = \frac{0}{3}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $3$ .

$\cos (x + \frac{π}{6}) = 0$

Step 2.2

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

$x + \frac{π}{6} = \arccos (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x + \frac{π}{6} = \frac{π}{2}$

Step 2.4

Move all terms not containing $x$ to the right side of the equation.

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Step 2.4.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$x = \frac{π}{2} - \frac{π}{6}$

Step 2.4.2

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

$x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Step 2.4.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.3.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$x = \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Step 2.4.3.2

Multiply $2$ by $3$ .

$x = \frac{π \cdot 3}{6} - \frac{π}{6}$

Step 2.4.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{6}$

Step 2.4.5

Simplify the numerator.

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Step 2.4.5.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{6}$

Step 2.4.5.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{6}$

Step 2.4.6

Cancel the common factor of $2$ and $6$ .

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Step 2.4.6.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{6}$

Step 2.4.6.2

Cancel the common factors.

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Step 2.4.6.2.1

Factor $2$ out of $6$ .

$x = \frac{2 π}{2 \cdot 3}$

Step 2.4.6.2.2

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 3}$

Step 2.4.6.2.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 2.5

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

$x + \frac{π}{6} = 2 π - \frac{π}{2}$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Simplify $2 π - \frac{π}{2}$ .

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Step 2.6.1.1

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

$x + \frac{π}{6} = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.6.1.2

Combine fractions.

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Step 2.6.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x + \frac{π}{6} = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.6.1.2.2

Combine the numerators over the common denominator.

$x + \frac{π}{6} = \frac{2 π \cdot 2 - π}{2}$

Step 2.6.1.3

Simplify the numerator.

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Step 2.6.1.3.1

Multiply $2$ by $2$ .

$x + \frac{π}{6} = \frac{4 π - π}{2}$

Step 2.6.1.3.2

Subtract $π$ from $4 π$ .

$x + \frac{π}{6} = \frac{3 π}{2}$

Step 2.6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.6.2.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$x = \frac{3 π}{2} - \frac{π}{6}$

Step 2.6.2.2

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

$x = \frac{3 π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Step 2.6.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.2.3.1

Multiply $\frac{3 π}{2}$ by $\frac{3}{3}$ .

$x = \frac{3 π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Step 2.6.2.3.2

Multiply $2$ by $3$ .

$x = \frac{3 π \cdot 3}{6} - \frac{π}{6}$

Step 2.6.2.4

Combine the numerators over the common denominator.

$x = \frac{3 π \cdot 3 - π}{6}$

Step 2.6.2.5

Simplify the numerator.

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Step 2.6.2.5.1

Multiply $3$ by $3$ .

$x = \frac{9 π - π}{6}$

Step 2.6.2.5.2

Subtract $π$ from $9 π$ .

$x = \frac{8 π}{6}$

Step 2.6.2.6

Cancel the common factor of $8$ and $6$ .

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Step 2.6.2.6.1

Factor $2$ out of $8 π$ .

$x = \frac{2 (4 π)}{6}$

Step 2.6.2.6.2

Cancel the common factors.

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Step 2.6.2.6.2.1

Factor $2$ out of $6$ .

$x = \frac{2 (4 π)}{2 (3)}$

Step 2.6.2.6.2.2

Cancel the common factor.

$x = \frac{2 (4 π)}{2 \cdot 3}$

Step 2.6.2.6.2.3

Rewrite the expression.

$x = \frac{4 π}{3}$

Step 2.7

Find the period of $\cos (x + \frac{π}{6})$ .

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Step 2.7.1

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

$\frac{2 π}{| b |}$

Step 2.7.2

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

$\frac{2 π}{| 1 |}$

Step 2.7.3

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

$\frac{2 π}{1}$

Step 2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.8

The period of the $\cos (x + \frac{π}{6})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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

Step 2.9

Consolidate the answers.

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

Step 3