Finite Math Examples

$f (x) = 5 \cos^{2} (π x)$

Step 1

Set $5 \cos^{2} (π x)$ equal to $0$ .

$5 \cos^{2} (π x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $5 \cos^{2} (π x) = 0$ by $5$ and simplify.

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Step 2.1.1

Divide each term in $5 \cos^{2} (π x) = 0$ by $5$ .

$\frac{5 \cos^{2} (π x)}{5} = \frac{0}{5}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $5$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{5 \cos^{2} (π x)}{5} = \frac{0}{5}$

Step 2.1.2.1.2

Divide $\cos^{2} (π x)$ by $1$ .

$\cos^{2} (π x) = \frac{0}{5}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $5$ .

$\cos^{2} (π x) = 0$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (π x) = \pm \sqrt{0}$

Step 2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.3.1

Rewrite $0$ as $0^{2}$ .

$\cos (π x) = \pm \sqrt{0^{2}}$

Step 2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (π x) = \pm 0$

Step 2.3.3

Plus or minus $0$ is $0$ .

$\cos (π x) = 0$

Step 2.4

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

$π x = \arccos (0)$

Step 2.5

Simplify the right side.

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Step 2.5.1

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

$π x = \frac{π}{2}$

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $π$ .

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Step 2.6.2.1.1

Cancel the common factor.

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

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{π}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{π}$

Step 2.6.3.2

Cancel the common factor of $π$ .

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Step 2.6.3.2.1

Cancel the common factor.

$x = \frac{π}{2} \cdot \frac{1}{π}$

Step 2.6.3.2.2

Rewrite the expression.

$x = \frac{1}{2}$

Step 2.7

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.

$π x = 2 π - \frac{π}{2}$

Step 2.8

Solve for $x$ .

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Step 2.8.1

Simplify.

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Step 2.8.1.1

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

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

Step 2.8.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 2.8.1.3

Combine the numerators over the common denominator.

$π x = \frac{2 π \cdot 2 - π}{2}$

Step 2.8.1.4

Multiply $2$ by $2$ .

$π x = \frac{4 π - π}{2}$

Step 2.8.1.5

Subtract $π$ from $4 π$ .

$π x = \frac{3 π}{2}$

Step 2.8.2

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

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Step 2.8.2.1

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

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

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $π$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

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

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{π}$

Step 2.8.2.3

Simplify the right side.

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Step 2.8.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{π}$

Step 2.8.2.3.2

Cancel the common factor of $π$ .

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Step 2.8.2.3.2.1

Factor $π$ out of $3 π$ .

$x = \frac{π \cdot 3}{2} \cdot \frac{1}{π}$

Step 2.8.2.3.2.2

Cancel the common factor.

$x = \frac{π \cdot 3}{2} \cdot \frac{1}{π}$

Step 2.8.2.3.2.3

Rewrite the expression.

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

Step 2.9

Find the period of $\cos (π x)$ .

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Step 2.9.1

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

$\frac{2 π}{| b |}$

Step 2.9.2

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

$\frac{2 π}{| π |}$

Step 2.9.3

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Step 2.9.4

Cancel the common factor of $π$ .

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Step 2.9.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 2.9.4.2

Divide $2$ by $1$ .

$2$

Step 2.10

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

$x = \frac{1}{2} + 2 n, \frac{3}{2} + 2 n$ , for any integer $n$

Step 2.11

Consolidate the answers.

$x = \frac{1}{2} + 1 n$ , for any integer $n$

Step 3