Finite Math Examples

$8 \cos (3 x) = \frac{1}{2}$

Step 1

Subtract $\frac{1}{2}$ from both sides of the equation.

$8 \cos (3 x) - \frac{1}{2} = 0$

Step 2

Add $\frac{1}{2}$ to both sides of the equation.

$8 \cos (3 x) = \frac{1}{2}$

Step 3

Divide each term in $8 \cos (3 x) = \frac{1}{2}$ by $8$ and simplify.

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Step 3.1

Divide each term in $8 \cos (3 x) = \frac{1}{2}$ by $8$ .

$\frac{8 \cos (3 x)}{8} = \frac{\frac{1}{2}}{8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{8 \cos (3 x)}{8} = \frac{\frac{1}{2}}{8}$

Step 3.2.1.2

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

$\cos (3 x) = \frac{\frac{1}{2}}{8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (3 x) = \frac{1}{2} \cdot \frac{1}{8}$

Step 3.3.2

Multiply $\frac{1}{2} \cdot \frac{1}{8}$ .

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Step 3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

$\cos (3 x) = \frac{1}{2 \cdot 8}$

Step 3.3.2.2

Multiply $2$ by $8$ .

$\cos (3 x) = \frac{1}{16}$

Step 4

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

$3 x = \arccos (\frac{1}{16})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arccos (\frac{1}{16})$ .

$3 x = 1.50825556$

Step 6

Divide each term in $3 x = 1.50825556$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 x = 1.50825556$ by $3$ .

$\frac{3 x}{3} = \frac{1.50825556}{3}$

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.

$\frac{3 x}{3} = \frac{1.50825556}{3}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.50825556}{3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $1.50825556$ by $3$ .

$x = 0.50275185$

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.

$3 x = 2 (3.14159265) - 1.50825556$

Step 8

Solve for $x$ .

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Step 8.1

Simplify.

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Step 8.1.1

Multiply $2$ by $3.14159265$ .

$3 x = 6.2831853 - 1.50825556$

Step 8.1.2

Subtract $1.50825556$ from $6.2831853$ .

$3 x = 4.77492974$

Step 8.2

Divide each term in $3 x = 4.77492974$ by $3$ and simplify.

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Step 8.2.1

Divide each term in $3 x = 4.77492974$ by $3$ .

$\frac{3 x}{3} = \frac{4.77492974}{3}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4.77492974}{3}$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.77492974}{3}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $4.77492974$ by $3$ .

$x = 1.59164324$

Step 9

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

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

$x = 0.50275185 + \frac{2 π n}{3}, 1.59164324 + \frac{2 π n}{3}$ , for any integer $n$