Algebra Examples

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

Step 1

Add $1$ to both sides of the equation.

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

Step 2

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

$\frac{2}{3} x - \frac{π}{2} = \arccos (1)$

Step 3

Simplify the left side.

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Step 3.1

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} - \frac{π}{2} = \arccos (1)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (1)$ is $0$ .

$\frac{2 x}{3} - \frac{π}{2} = 0$

Step 5

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

$\frac{2 x}{3} = \frac{π}{2}$

Step 6

Multiply both sides of the equation by $\frac{3}{2}$ .

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

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

Cancel the common factor.

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

Step 7.1.1.1.2

Rewrite the expression.

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

Step 7.1.1.2

Cancel the common factor of $2$ .

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Step 7.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 7.1.1.2.2

Cancel the common factor.

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

Step 7.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} \cdot \frac{π}{2}$

Step 7.2

Simplify the right side.

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Step 7.2.1

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

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Step 7.2.1.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 7.2.1.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 8

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.

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

Step 9

Solve for $x$ .

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Step 9.1

Subtract $0$ from $2 π$ .

$\frac{2 x}{3} - \frac{π}{2} = 2 π$

Step 9.2

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

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Step 9.2.1

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

$\frac{2 x}{3} = 2 π + \frac{π}{2}$

Step 9.2.2

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

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

Step 9.2.3

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 9.2.4

Combine the numerators over the common denominator.

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

Step 9.2.5

Simplify the numerator.

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Step 9.2.5.1

Multiply $2$ by $2$ .

$\frac{2 x}{3} = \frac{4 π + π}{2}$

Step 9.2.5.2

Add $4 π$ and $π$ .

$\frac{2 x}{3} = \frac{5 π}{2}$

Step 9.3

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 x}{3} = \frac{3}{2} \cdot \frac{5 π}{2}$

Step 9.4

Simplify both sides of the equation.

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Step 9.4.1

Simplify the left side.

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Step 9.4.1.1

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 9.4.1.1.1

Cancel the common factor of $3$ .

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Step 9.4.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 x}{3} = \frac{3}{2} \cdot \frac{5 π}{2}$

Step 9.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 x) = \frac{3}{2} \cdot \frac{5 π}{2}$

Step 9.4.1.1.2

Cancel the common factor of $2$ .

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Step 9.4.1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{3}{2} \cdot \frac{5 π}{2}$

Step 9.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{3}{2} \cdot \frac{5 π}{2}$

Step 9.4.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} \cdot \frac{5 π}{2}$

Step 9.4.2

Simplify the right side.

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Step 9.4.2.1

Multiply $\frac{3}{2} \cdot \frac{5 π}{2}$ .

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Step 9.4.2.1.1

Multiply $\frac{3}{2}$ by $\frac{5 π}{2}$ .

$x = \frac{3 (5 π)}{2 \cdot 2}$

Step 9.4.2.1.2

Multiply $5$ by $3$ .

$x = \frac{15 π}{2 \cdot 2}$

Step 9.4.2.1.3

Multiply $2$ by $2$ .

$x = \frac{15 π}{4}$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $\frac{2}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2}{3} |}$

Step 10.3

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Step 10.5

Cancel the common factor of $2$ .

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Step 10.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Step 10.5.2

Cancel the common factor.

$2 π \frac{3}{2}$

Step 10.5.3

Rewrite the expression.

$π \cdot 3$

Step 10.6

Move $3$ to the left of $π$ .

$3 π$

Step 11

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

$x = \frac{3 π}{4} + 3 π n, \frac{15 π}{4} + 3 π n$ , for any integer $n$

Step 12

Consolidate the answers.

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