Precalculus Examples

2 {cos}^{2} (x) - 1 = 0

$2 \cos^{2} (x) - 1 = 0$

Step 1

Add

1

$1$ to both sides of the equation.

2 {cos}^{2} (x) = 1

$2 \cos^{2} (x) = 1$

Step 2

Divide each term in

2 {cos}^{2} (x) = 1

$2 \cos^{2} (x) = 1$ by

2

$2$ and simplify.

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Step 2.1

Divide each term in

2 {cos}^{2} (x) = 1

$2 \cos^{2} (x) = 1$ by

2

$2$ .

\frac{2 {cos}^{2} (x)}{2} = \frac{1}{2}

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide

$\cos^{2} (x)$ by

$1$ .

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

Step 3

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

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

Step 4

Simplify

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

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Step 4.1

Rewrite

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

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

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

Step 4.2

Any root of

$1$ is

$1$ .

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

Step 4.3

Multiply

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

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply

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

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 4.4.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 4.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.4.5

Add

$1$ and

$1$ .

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

Step 4.4.6

Rewrite

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

$2$ .

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Step 4.4.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

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

Step 4.4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 4.4.6.4

Cancel the common factor of

$2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

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

Step 4.4.6.5

Evaluate the exponent.

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

Step 5

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

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Step 5.1

First, use the positive value of the

$\pm$ to find the first solution.

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 5.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 5.3

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

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

Step 6

Set up each of the solutions to solve for

$x$ .

$\cos (x) = \frac{\sqrt{2}}{2}$

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

Step 7

Solve for

$x$ in

$\cos (x) = \frac{\sqrt{2}}{2}$ .

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Step 7.1

Take the inverse cosine of both sides of the equation to extract

$x$ from inside the cosine.

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of

$\arccos (\frac{\sqrt{2}}{2})$ is

$\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 7.3

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.

$x = 2 π - \frac{π}{4}$

Step 7.4

Simplify

$2 π - \frac{π}{4}$ .

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Step 7.4.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 7.4.2

Combine fractions.

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Step 7.4.2.1

Combine

$2 π$ and

$\frac{4}{4}$ .

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

Step 7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 7.4.3

Simplify the numerator.

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Step 7.4.3.1

Multiply

$4$ by

$2$ .

$x = \frac{8 π - π}{4}$

Step 7.4.3.2

Subtract

$π$ from

$8 π$ .

$x = \frac{7 π}{4}$

Step 7.5

Find the period of

$\cos (x)$ .

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Step 7.5.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.5.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 7.5.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 7.6

The period of the

$\cos (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

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

$n$

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

$n$

Step 8

Solve for

$x$ in

$\cos (x) = - \frac{\sqrt{2}}{2}$ .

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Step 8.1

Take the inverse cosine of both sides of the equation to extract

$x$ from inside the cosine.

$x = \arccos (- \frac{\sqrt{2}}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of

$\arccos (- \frac{\sqrt{2}}{2})$ is

$\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 8.3

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.

$x = 2 π - \frac{3 π}{4}$

Step 8.4

Simplify

$2 π - \frac{3 π}{4}$ .

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Step 8.4.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 8.4.2

Combine fractions.

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Step 8.4.2.1

Combine

$2 π$ and

$\frac{4}{4}$ .

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

Step 8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 8.4.3

Simplify the numerator.

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Step 8.4.3.1

Multiply

$4$ by

$2$ .

$x = \frac{8 π - 3 π}{4}$

Step 8.4.3.2

Subtract

$3 π$ from

$8 π$ .

$x = \frac{5 π}{4}$

Step 8.5

Find the period of

$\cos (x)$ .

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Step 8.5.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.5.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 8.5.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 8.6

The period of the

$\cos (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

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

$n$

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

$n$

Step 9

List all of the solutions.

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

$n$

Step 10

Consolidate the answers.

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

$n$