Precalculus Examples

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

Step 1

Set $\frac{1}{2} \cdot \cos (2 x)$ equal to $0$ .

$\frac{1}{2} \cdot \cos (2 x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cos (2 x)) = 2 \cdot 0$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $2 (\frac{1}{2} \cos (2 x))$ .

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Step 2.2.1.1.1

Combine $\frac{1}{2}$ and $\cos (2 x)$ .

$2 \frac{\cos (2 x)}{2} = 2 \cdot 0$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$2 \frac{\cos (2 x)}{2} = 2 \cdot 0$

Step 2.2.1.1.2.2

Rewrite the expression.

$\cos (2 x) = 2 \cdot 0$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $2$ by $0$ .

$\cos (2 x) = 0$

Step 2.3

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

$2 x = \arccos (0)$

Step 2.4

Simplify the right side.

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Step 2.4.1

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

$2 x = \frac{π}{2}$

Step 2.5

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

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Step 2.5.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.5.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.5.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.5.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 2.6

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.

$2 x = 2 π - \frac{π}{2}$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Simplify.

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Step 2.7.1.1

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

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

Step 2.7.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 2.7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 2.7.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 2.7.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 2.7.2

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

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Step 2.7.2.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

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

Step 2.7.2.2

Simplify the left side.

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Step 2.7.2.2.1

Cancel the common factor of $2$ .

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Step 2.7.2.2.1.1

Cancel the common factor.

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

Step 2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 2.7.2.3

Simplify the right side.

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Step 2.7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 2.7.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 2.7.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 2.7.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 2.8

Find the period of $\cos (2 x)$ .

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Step 2.8.1

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

$\frac{2 π}{| b |}$

Step 2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.8.4

Cancel the common factor of $2$ .

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Step 2.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.8.4.2

Divide $π$ by $1$ .

$π$

Step 2.9

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

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

Step 2.10

Consolidate the answers.

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

Step 3