Algebra Examples

$35 = 12 \cos (66 π x) + 40$

Step 1

Rewrite the equation as $12 \cos (66 π x) + 40 = 35$ .

$12 \cos (66 π x) + 40 = 35$

Step 2

Move all terms not containing $\cos (66 π x)$ to the right side of the equation.

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Step 2.1

Subtract $40$ from both sides of the equation.

$12 \cos (66 π x) = 35 - 40$

Step 2.2

Subtract $40$ from $35$ .

$12 \cos (66 π x) = - 5$

Step 3

Divide each term in $12 \cos (66 π x) = - 5$ by $12$ and simplify.

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Step 3.1

Divide each term in $12 \cos (66 π x) = - 5$ by $12$ .

$\frac{12 \cos (66 π x)}{12} = \frac{- 5}{12}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $12$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{12 \cos (66 π x)}{12} = \frac{- 5}{12}$

Step 3.2.1.2

Divide $\cos (66 π x)$ by $1$ .

$\cos (66 π x) = \frac{- 5}{12}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$\cos (66 π x) = - \frac{5}{12}$

Step 4

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

$66 π x = \arccos (- \frac{5}{12})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arccos (- \frac{5}{12})$ .

$66 π x = 2.00057175$

Step 6

Divide each term in $66 π x = 2.00057175$ by $66 π$ and simplify.

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Step 6.1

Divide each term in $66 π x = 2.00057175$ by $66 π$ .

$\frac{66 π x}{66 π} = \frac{2.00057175}{66 π}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $66$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{66 π x}{66 π} = \frac{2.00057175}{66 π}$

Step 6.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{2.00057175}{66 π}$

Step 6.2.2

Cancel the common factor of $π$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{2.00057175}{66 π}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{2.00057175}{66 π}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Replace $π$ with an approximation.

$x = \frac{2.00057175}{66 \cdot 3.14159265}$

Step 6.3.2

Multiply $66$ by $3.14159265$ .

$x = \frac{2.00057175}{207.34511513}$

Step 6.3.3

Divide $2.00057175$ by $207.34511513$ .

$x = 0.00964851$

Step 7

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$66 π x = 2 (3.14159265) - 2.00057175$

Step 8

Solve for $x$ .

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Step 8.1

Simplify.

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Step 8.1.1

Multiply $2$ by $3.14159265$ .

$66 π x = 6.2831853 - 2.00057175$

Step 8.1.2

Subtract $2.00057175$ from $6.2831853$ .

$66 π x = 4.28261354$

Step 8.2

Divide each term in $66 π x = 4.28261354$ by $66 π$ and simplify.

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Step 8.2.1

Divide each term in $66 π x = 4.28261354$ by $66 π$ .

$\frac{66 π x}{66 π} = \frac{4.28261354}{66 π}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $66$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{66 π x}{66 π} = \frac{4.28261354}{66 π}$

Step 8.2.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{4.28261354}{66 π}$

Step 8.2.2.2

Cancel the common factor of $π$ .

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Step 8.2.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{4.28261354}{66 π}$

Step 8.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{4.28261354}{66 π}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Replace $π$ with an approximation.

$x = \frac{4.28261354}{66 \cdot 3.14159265}$

Step 8.2.3.2

Multiply $66$ by $3.14159265$ .

$x = \frac{4.28261354}{207.34511513}$

Step 8.2.3.3

Divide $4.28261354$ by $207.34511513$ .

$x = 0.02065451$

Step 9

Find the period of $\cos (66 π x)$ .

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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 $66 π$ in the formula for period.

$\frac{2 π}{| 66 π |}$

Step 9.3

$66 π$ is approximately $207.34511513$ which is positive so remove the absolute value

$\frac{2 π}{66 π}$

Step 9.4

Cancel the common factor of $2$ and $66$ .

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Step 9.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{66 π}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $2$ out of $66 π$ .

$\frac{2 (π)}{2 (33 π)}$

Step 9.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (33 π)}$

Step 9.4.2.3

Rewrite the expression.

$\frac{π}{33 π}$

Step 9.5

Cancel the common factor of $π$ .

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Step 9.5.1

Cancel the common factor.

$\frac{π}{33 π}$

Step 9.5.2

Rewrite the expression.

$\frac{1}{33}$

Step 10

The period of the $\cos (66 π x)$ function is $\frac{1}{33}$ so values will repeat every $\frac{1}{33}$ radians in both directions.

$x = 0.00964851 + \frac{1 n}{33}, 0.02065451 + \frac{1 n}{33}$ , for any integer $n$