Finite Math Examples

$y = 4 \cos (3 x + \frac{π}{2})$

Step 1

Set $4 \cos (3 x + \frac{π}{2})$ equal to $0$ .

$4 \cos (3 x + \frac{π}{2}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $4 \cos (3 x + \frac{π}{2}) = 0$ by $4$ and simplify.

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Step 2.1.1

Divide each term in $4 \cos (3 x + \frac{π}{2}) = 0$ by $4$ .

$\frac{4 \cos (3 x + \frac{π}{2})}{4} = \frac{0}{4}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{4 \cos (3 x + \frac{π}{2})}{4} = \frac{0}{4}$

Step 2.1.2.1.2

Divide $\cos (3 x + \frac{π}{2})$ by $1$ .

$\cos (3 x + \frac{π}{2}) = \frac{0}{4}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $4$ .

$\cos (3 x + \frac{π}{2}) = 0$

Step 2.2

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

$3 x + \frac{π}{2} = \arccos (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$3 x + \frac{π}{2} = \frac{π}{2}$

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 $\frac{π}{2}$ from both sides of the equation.

$3 x = \frac{π}{2} - \frac{π}{2}$

Step 2.4.2

Combine the numerators over the common denominator.

$3 x = \frac{π - π}{2}$

Step 2.4.3

Subtract $π$ from $π$ .

$3 x = \frac{0}{2}$

Step 2.4.4

Divide $0$ by $2$ .

$3 x = 0$

Step 2.5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 2.5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $3$ .

$x = 0$

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.

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

Step 2.7

Solve for $x$ .

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Step 2.7.1

Simplify $2 π - \frac{π}{2}$ .

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Step 2.7.1.1

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

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

Step 2.7.1.2

Combine fractions.

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Step 2.7.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 2.7.1.2.2

Combine the numerators over the common denominator.

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

Step 2.7.1.3

Simplify the numerator.

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Step 2.7.1.3.1

Multiply $2$ by $2$ .

$3 x + \frac{π}{2} = \frac{4 π - π}{2}$

Step 2.7.1.3.2

Subtract $π$ from $4 π$ .

$3 x + \frac{π}{2} = \frac{3 π}{2}$

Step 2.7.2

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

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Step 2.7.2.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$3 x = \frac{3 π}{2} - \frac{π}{2}$

Step 2.7.2.2

Combine the numerators over the common denominator.

$3 x = \frac{3 π - π}{2}$

Step 2.7.2.3

Subtract $π$ from $3 π$ .

$3 x = \frac{2 π}{2}$

Step 2.7.2.4

Cancel the common factor of $2$ .

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Step 2.7.2.4.1

Cancel the common factor.

$3 x = \frac{2 π}{2}$

Step 2.7.2.4.2

Divide $π$ by $1$ .

$3 x = π$

Step 2.7.3

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

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Step 2.7.3.1

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

$\frac{3 x}{3} = \frac{π}{3}$

Step 2.7.3.2

Simplify the left side.

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Step 2.7.3.2.1

Cancel the common factor of $3$ .

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Step 2.7.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{π}{3}$

Step 2.7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{3}$

Step 2.8

Find the period of $\cos (3 x + \frac{π}{2})$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 2.8.3

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

$\frac{2 π}{3}$

Step 2.9

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

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

Step 2.10

Consolidate the answers.

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

Step 3