Precalculus Examples

$\cos (3 x) = \frac{\sqrt{3}}{2}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $2$ .

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

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$\cos (3 x) \cdot 2$

Step 3

Move $2$ to the left of $\cos (3 x)$ .

$\frac{2 \cos (3 x)}{2} = \frac{\sqrt{3}}{2}$

Step 4

Simplify $\cos (3 x) \cdot \frac{2}{2}$ .

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Step 4.1

Divide $2$ by $2$ .

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

Step 4.2

Multiply $\cos (3 x)$ by $1$ .

$\cos (3 x) = \frac{\sqrt{3}}{2}$

Step 5

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

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

Step 6

Simplify the right side.

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Step 6.1

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

$3 x = \frac{π}{6}$

Step 7

Divide each term in $3 x = \frac{π}{6}$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 x = \frac{π}{6}$ by $3$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{3}$

Step 7.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{3}$ .

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Step 7.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{3}$ .

$x = \frac{π}{6 \cdot 3}$

Step 7.3.2.2

Multiply $6$ by $3$ .

$x = \frac{π}{18}$

Step 8

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.

$3 x = 2 π - \frac{π}{6}$

Step 9

Solve for $x$ .

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 9.1.3

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 6 - π}{6}$

Step 9.1.4

Multiply $6$ by $2$ .

$3 x = \frac{12 π - π}{6}$

Step 9.1.5

Subtract $π$ from $12 π$ .

$3 x = \frac{11 π}{6}$

Step 9.2

Divide each term in $3 x = \frac{11 π}{6}$ by $3$ and simplify.

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Step 9.2.1

Divide each term in $3 x = \frac{11 π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{11 π}{6}}{3}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $3$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{11 π}{6}}{3}$

Step 9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{11 π}{6}}{3}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{11 π}{6} \cdot \frac{1}{3}$

Step 9.2.3.2

Multiply $\frac{11 π}{6} \cdot \frac{1}{3}$ .

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Step 9.2.3.2.1

Multiply $\frac{11 π}{6}$ by $\frac{1}{3}$ .

$x = \frac{11 π}{6 \cdot 3}$

Step 9.2.3.2.2

Multiply $6$ by $3$ .

$x = \frac{11 π}{18}$

Step 10

Find the period of $\cos (3 x)$ .

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 10.3

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

$\frac{2 π}{3}$

Step 11

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

$x = \frac{π}{18} + \frac{2 π n}{3}, \frac{11 π}{18} + \frac{2 π n}{3}$ , for any integer $n$