Precalculus Examples

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

Step 1

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

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

Step 2

Any root of $1$ is $1$ .

$\cos (x) = \pm 1$

Step 3

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

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Step 3.1

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

$\cos (x) = 1$

Step 3.2

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

$\cos (x) = - 1$

Step 3.3

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

$\cos (x) = 1, - 1$

Step 4

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

$\cos (x) = 1$

$\cos (x) = - 1$

Step 5

Solve for $x$ in $\cos (x) = 1$ .

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Step 5.1

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

$x = \arccos (1)$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

$x = 0$

Step 5.3

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.

$x = 2 π - 0$

Step 5.4

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 5.5

Find the period of $\cos (x)$ .

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Step 5.5.1

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

$\frac{2 π}{| b |}$

Step 5.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.5.3

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

$\frac{2 π}{1}$

Step 5.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.6

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

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 6

Solve for $x$ in $\cos (x) = - 1$ .

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Step 6.1

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

$x = \arccos (- 1)$

Step 6.2

Simplify the right side.

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Step 6.2.1

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

$x = π$

Step 6.3

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.

$x = 2 π - π$

Step 6.4

Subtract $π$ from $2 π$ .

$x = π$

Step 6.5

Find the period of $\cos (x)$ .

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Step 6.5.1

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

$\frac{2 π}{| b |}$

Step 6.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.5.3

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

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

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

$x = π + 2 π n$ , for any integer $n$

Step 7

List all of the solutions.

$x = 2 π n, 2 π + 2 π n, π + 2 π n$ , for any integer $n$

Step 8

Consolidate the answers.

$x = π n$ , for any integer $n$