Algebra Examples

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

Step 1

Add $1$ to both sides of the equation.

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

Step 2

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

$\cos (\frac{x}{2} + \frac{π}{3}) = \pm \sqrt{1}$

Step 3

Any root of $1$ is $1$ .

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

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 5

Set up each of the solutions to solve for $x$ .

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

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

Step 6

Solve for $x$ in $\cos (\frac{x}{2} + \frac{π}{3}) = 1$ .

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

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

$\frac{x}{2} + \frac{π}{3} = 0$

Step 6.3

Subtract $\frac{π}{3}$ from both sides of the equation.

$\frac{x}{2} = - \frac{π}{3}$

Step 6.4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{π}{3})$

Step 6.5

Simplify both sides of the equation.

Tap for more steps...

Step 6.5.1

Simplify the left side.

Tap for more steps...

Step 6.5.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{π}{3})$

Step 6.5.1.1.2

Rewrite the expression.

$x = 2 (- \frac{π}{3})$

Step 6.5.2

Simplify the right side.

Tap for more steps...

Step 6.5.2.1

Simplify $2 (- \frac{π}{3})$ .

Tap for more steps...

Step 6.5.2.1.1

Multiply $2 (- \frac{π}{3})$ .

Tap for more steps...

Step 6.5.2.1.1.1

Multiply $- 1$ by $2$ .

$x = - 2 \frac{π}{3}$

Step 6.5.2.1.1.2

Combine $- 2$ and $\frac{π}{3}$ .

$x = \frac{- 2 π}{3}$

Step 6.5.2.1.2

Move the negative in front of the fraction.

$x = - \frac{2 π}{3}$

Step 6.6

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.

$\frac{x}{2} + \frac{π}{3} = 2 π - 0$

Step 6.7

Solve for $x$ .

Tap for more steps...

Step 6.7.1

Subtract $0$ from $2 π$ .

$\frac{x}{2} + \frac{π}{3} = 2 π$

Step 6.7.2

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

Tap for more steps...

Step 6.7.2.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$\frac{x}{2} = 2 π - \frac{π}{3}$

Step 6.7.2.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.7.2.3

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 6.7.2.4

Combine the numerators over the common denominator.

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

Step 6.7.2.5

Simplify the numerator.

Tap for more steps...

Step 6.7.2.5.1

Multiply $3$ by $2$ .

$\frac{x}{2} = \frac{6 π - π}{3}$

Step 6.7.2.5.2

Subtract $π$ from $6 π$ .

$\frac{x}{2} = \frac{5 π}{3}$

Step 6.7.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{5 π}{3}$

Step 6.7.4

Simplify both sides of the equation.

Tap for more steps...

Step 6.7.4.1

Simplify the left side.

Tap for more steps...

Step 6.7.4.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.7.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{5 π}{3}$

Step 6.7.4.1.1.2

Rewrite the expression.

$x = 2 \frac{5 π}{3}$

Step 6.7.4.2

Simplify the right side.

Tap for more steps...

Step 6.7.4.2.1

Multiply $2 \frac{5 π}{3}$ .

Tap for more steps...

Step 6.7.4.2.1.1

Combine $2$ and $\frac{5 π}{3}$ .

$x = \frac{2 (5 π)}{3}$

Step 6.7.4.2.1.2

Multiply $5$ by $2$ .

$x = \frac{10 π}{3}$

Step 6.8

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

Tap for more steps...

Step 6.8.1

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

$\frac{2 π}{| b |}$

Step 6.8.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 6.8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 6.8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 6.8.5

Multiply $2$ by $2$ .

$4 π$

Step 6.9

Add $4 π$ to every negative angle to get positive angles.

Tap for more steps...

Step 6.9.1

Add $4 π$ to $- \frac{2 π}{3}$ to find the positive angle.

$- \frac{2 π}{3} + 4 π$

Step 6.9.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 6.9.3

Combine fractions.

Tap for more steps...

Step 6.9.3.1

Combine $4 π$ and $\frac{3}{3}$ .

$\frac{4 π \cdot 3}{3} - \frac{2 π}{3}$

Step 6.9.3.2

Combine the numerators over the common denominator.

$\frac{4 π \cdot 3 - 2 π}{3}$

Step 6.9.4

Simplify the numerator.

Tap for more steps...

Step 6.9.4.1

Multiply $3$ by $4$ .

$\frac{12 π - 2 π}{3}$

Step 6.9.4.2

Subtract $2 π$ from $12 π$ .

$\frac{10 π}{3}$

Step 6.9.5

List the new angles.

$x = \frac{10 π}{3}$

Step 6.10

The period of the $\cos (\frac{x}{2} + \frac{π}{3})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{10 π}{3} + 4 π n, \frac{10 π}{3} + 4 π n$ , for any integer $n$

Step 7

Solve for $x$ in $\cos (\frac{x}{2} + \frac{π}{3}) = - 1$ .

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

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

$\frac{x}{2} + \frac{π}{3} = π$

Step 7.3

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

Tap for more steps...

Step 7.3.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$\frac{x}{2} = π - \frac{π}{3}$

Step 7.3.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.3.3

Combine $π$ and $\frac{3}{3}$ .

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

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Simplify the numerator.

Tap for more steps...

Step 7.3.5.1

Move $3$ to the left of $π$ .

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

Step 7.3.5.2

Subtract $π$ from $3 π$ .

$\frac{x}{2} = \frac{2 π}{3}$

Step 7.4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{2 π}{3}$

Step 7.5

Simplify both sides of the equation.

Tap for more steps...

Step 7.5.1

Simplify the left side.

Tap for more steps...

Step 7.5.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{2 π}{3}$

Step 7.5.1.1.2

Rewrite the expression.

$x = 2 \frac{2 π}{3}$

Step 7.5.2

Simplify the right side.

Tap for more steps...

Step 7.5.2.1

Multiply $2 \frac{2 π}{3}$ .

Tap for more steps...

Step 7.5.2.1.1

Combine $2$ and $\frac{2 π}{3}$ .

$x = \frac{2 (2 π)}{3}$

Step 7.5.2.1.2

Multiply $2$ by $2$ .

$x = \frac{4 π}{3}$

Step 7.6

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.

$\frac{x}{2} + \frac{π}{3} = 2 π - π$

Step 7.7

Solve for $x$ .

Tap for more steps...

Step 7.7.1

Subtract $π$ from $2 π$ .

$\frac{x}{2} + \frac{π}{3} = π$

Step 7.7.2

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

Tap for more steps...

Step 7.7.2.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$\frac{x}{2} = π - \frac{π}{3}$

Step 7.7.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.7.2.3

Combine $π$ and $\frac{3}{3}$ .

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

Step 7.7.2.4

Combine the numerators over the common denominator.

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

Step 7.7.2.5

Simplify the numerator.

Tap for more steps...

Step 7.7.2.5.1

Move $3$ to the left of $π$ .

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

Step 7.7.2.5.2

Subtract $π$ from $3 π$ .

$\frac{x}{2} = \frac{2 π}{3}$

Step 7.7.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{2 π}{3}$

Step 7.7.4

Simplify both sides of the equation.

Tap for more steps...

Step 7.7.4.1

Simplify the left side.

Tap for more steps...

Step 7.7.4.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.7.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{2 π}{3}$

Step 7.7.4.1.1.2

Rewrite the expression.

$x = 2 \frac{2 π}{3}$

Step 7.7.4.2

Simplify the right side.

Tap for more steps...

Step 7.7.4.2.1

Multiply $2 \frac{2 π}{3}$ .

Tap for more steps...

Step 7.7.4.2.1.1

Combine $2$ and $\frac{2 π}{3}$ .

$x = \frac{2 (2 π)}{3}$

Step 7.7.4.2.1.2

Multiply $2$ by $2$ .

$x = \frac{4 π}{3}$

Step 7.8

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

Tap for more steps...

Step 7.8.1

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

$\frac{2 π}{| b |}$

Step 7.8.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 7.8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 7.8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 7.8.5

Multiply $2$ by $2$ .

$4 π$

Step 7.9

The period of the $\cos (\frac{x}{2} + \frac{π}{3})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

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

Step 8

List all of the solutions.

$x = \frac{10 π}{3} + 4 π n, \frac{10 π}{3} + 4 π n, \frac{4 π}{3} + 4 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{4 π}{3} + 2 π n$ , for any integer $n$