Algebra Examples

$\cos^{2} (θ) = \frac{1}{2}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $2$ .

$\cos^{2} (θ) \cdot \frac{2}{2} = \frac{1}{2}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$\cos^{2} (θ) \cdot 2$

Step 3

Move $2$ to the left of $\cos^{2} (θ)$ .

$\frac{2 \cos^{2} (θ)}{2} = \frac{1}{2}$

Step 4

Simplify $\cos^{2} (θ) \cdot \frac{2}{2}$ .

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Step 4.1

Divide $2$ by $2$ .

$\cos^{2} (θ) \cdot 1 = \frac{1}{2}$

Step 4.2

Multiply $\cos^{2} (θ)$ by $1$ .

$\cos^{2} (θ) = \frac{1}{2}$

Step 5

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

$\cos (θ) = \pm \sqrt{\frac{1}{2}}$

Step 6

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\cos (θ) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 6.2

Any root of $1$ is $1$ .

$\cos (θ) = \pm \frac{1}{\sqrt{2}}$

Step 6.3

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

$\cos (θ) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

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

$\cos (θ) = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.4.5

Add $1$ and $1$ .

$\cos (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos (θ) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (θ) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\cos (θ) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.4.6.4

Cancel the common factor of $2$ .

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Step 6.4.6.4.1

Cancel the common factor.

$\cos (θ) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.4.6.4.2

Rewrite the expression.

$\cos (θ) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 6.4.6.5

Evaluate the exponent.

$\cos (θ) = \pm \frac{\sqrt{2}}{2}$

Step 7

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

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Step 7.1

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

$\cos (θ) = \frac{\sqrt{2}}{2}$

Step 7.2

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

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Step 7.3

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

$\cos (θ) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 8

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

$\cos (θ) = \frac{\sqrt{2}}{2}$

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Step 9

Solve for $θ$ in $\cos (θ) = \frac{\sqrt{2}}{2}$ .

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Step 9.1

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

$θ = \arccos (\frac{\sqrt{2}}{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$θ = \frac{π}{4}$

Step 9.3

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.

$θ = 2 π - \frac{π}{4}$

Step 9.4

Simplify $2 π - \frac{π}{4}$ .

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Step 9.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$θ = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 9.4.2

Combine fractions.

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Step 9.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$θ = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 9.4.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 4 - π}{4}$

Step 9.4.3

Simplify the numerator.

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Step 9.4.3.1

Multiply $4$ by $2$ .

$θ = \frac{8 π - π}{4}$

Step 9.4.3.2

Subtract $π$ from $8 π$ .

$θ = \frac{7 π}{4}$

Step 9.5

Find the period of $\cos (θ)$ .

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Step 9.5.1

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

$\frac{2 π}{| b |}$

Step 9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

The period of the $\cos (θ)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 10

Solve for $θ$ in $\cos (θ) = - \frac{\sqrt{2}}{2}$ .

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Step 10.1

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

$θ = \arccos (- \frac{\sqrt{2}}{2})$

Step 10.2

Simplify the right side.

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Step 10.2.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$θ = \frac{3 π}{4}$

Step 10.3

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.

$θ = 2 π - \frac{3 π}{4}$

Step 10.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 10.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 10.4.2

Combine fractions.

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Step 10.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 10.4.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 4 - 3 π}{4}$

Step 10.4.3

Simplify the numerator.

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Step 10.4.3.1

Multiply $4$ by $2$ .

$θ = \frac{8 π - 3 π}{4}$

Step 10.4.3.2

Subtract $3 π$ from $8 π$ .

$θ = \frac{5 π}{4}$

Step 10.5

Find the period of $\cos (θ)$ .

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Step 10.5.1

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

$\frac{2 π}{| b |}$

Step 10.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 10.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

The period of the $\cos (θ)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 11

List all of the solutions.

$θ = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 12

Consolidate the answers.

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