Trigonometry Examples

$2 \cos (3 x - \frac{1}{6} \cdot π) = \sqrt{3}$

Step 1

Divide each term in $2 \cos (3 x - \frac{1}{6} \cdot π) = \sqrt{3}$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \cos (3 x - \frac{1}{6} \cdot π) = \sqrt{3}$ by $2$ .

$\frac{2 \cos (3 x - \frac{1}{6} \cdot π)}{2} = \frac{\sqrt{3}}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cos (3 x - \frac{1}{6} \cdot π)}{2} = \frac{\sqrt{3}}{2}$

Step 1.2.1.2

Divide $\cos (3 x - \frac{1}{6} \cdot π)$ by $1$ .

$\cos (3 x - \frac{1}{6} \cdot π) = \frac{\sqrt{3}}{2}$

Step 2

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

$3 x - \frac{1}{6} \cdot π = \arccos (\frac{\sqrt{3}}{2})$

Step 3

Simplify the left side.

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Step 3.1

Combine $π$ and $\frac{1}{6}$ .

$3 x - \frac{π}{6} = \arccos (\frac{\sqrt{3}}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$3 x - \frac{π}{6} = \frac{π}{6}$

Step 5

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

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Step 5.1

Add $\frac{π}{6}$ to both sides of the equation.

$3 x = \frac{π}{6} + \frac{π}{6}$

Step 5.2

Combine the numerators over the common denominator.

$3 x = \frac{π + π}{6}$

Step 5.3

Add $π$ and $π$ .

$3 x = \frac{2 π}{6}$

Step 5.4

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

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Step 5.4.1

Factor $2$ out of $2 π$ .

$3 x = \frac{2 (π)}{6}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $2$ out of $6$ .

$3 x = \frac{2 π}{2 \cdot 3}$

Step 5.4.2.2

Cancel the common factor.

$3 x = \frac{2 π}{2 \cdot 3}$

Step 5.4.2.3

Rewrite the expression.

$3 x = \frac{π}{3}$

Step 6

Divide each term in $3 x = \frac{π}{3}$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 x = \frac{π}{3}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{3}}{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{3}}{3}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{3}$

Step 6.3.2

Multiply $\frac{π}{3} \cdot \frac{1}{3}$ .

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Step 6.3.2.1

Multiply $\frac{π}{3}$ by $\frac{1}{3}$ .

$x = \frac{π}{3 \cdot 3}$

Step 6.3.2.2

Multiply $3$ by $3$ .

$x = \frac{π}{9}$

Step 7

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.

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

Step 8

Solve for $x$ .

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Step 8.1

Simplify $2 π - \frac{π}{6}$ .

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Step 8.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 8.1.2

Combine fractions.

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Step 8.1.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 8.1.2.2

Combine the numerators over the common denominator.

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

Step 8.1.3

Simplify the numerator.

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Step 8.1.3.1

Multiply $6$ by $2$ .

$3 x - \frac{π}{6} = \frac{12 π - π}{6}$

Step 8.1.3.2

Subtract $π$ from $12 π$ .

$3 x - \frac{π}{6} = \frac{11 π}{6}$

Step 8.2

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

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Step 8.2.1

Add $\frac{π}{6}$ to both sides of the equation.

$3 x = \frac{11 π}{6} + \frac{π}{6}$

Step 8.2.2

Combine the numerators over the common denominator.

$3 x = \frac{11 π + π}{6}$

Step 8.2.3

Add $11 π$ and $π$ .

$3 x = \frac{12 π}{6}$

Step 8.2.4

Cancel the common factor of $12$ and $6$ .

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Step 8.2.4.1

Factor $6$ out of $12 π$ .

$3 x = \frac{6 (2 π)}{6}$

Step 8.2.4.2

Cancel the common factors.

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Step 8.2.4.2.1

Factor $6$ out of $6$ .

$3 x = \frac{6 (2 π)}{6 (1)}$

Step 8.2.4.2.2

Cancel the common factor.

$3 x = \frac{6 (2 π)}{6 \cdot 1}$

Step 8.2.4.2.3

Rewrite the expression.

$3 x = \frac{2 π}{1}$

Step 8.2.4.2.4

Divide $2 π$ by $1$ .

$3 x = 2 π$

Step 8.3

Divide each term in $3 x = 2 π$ by $3$ and simplify.

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Step 8.3.1

Divide each term in $3 x = 2 π$ by $3$ .

$\frac{3 x}{3} = \frac{2 π}{3}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $3$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2 π}{3}$

Step 8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{3}$

Step 9

Find the period of $\cos (3 x - \frac{1}{6} \cdot π)$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 10

The period of the $\cos (3 x - \frac{1}{6} \cdot π)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{π}{9} + \frac{2 π n}{3}, \frac{2 π}{3} + \frac{2 π n}{3}$ , for any integer $n$