Precalculus Examples

$2 \cos (2 x) + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$2 \cos (2 x) = - 1$

Step 2

Divide each term in $2 \cos (2 x) = - 1$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \cos (2 x) = - 1$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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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 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$2 x = \frac{2 π}{3}$

Step 5

Divide each term in $2 x = \frac{2 π}{3}$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = \frac{2 π}{3}$ by $2$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{2 π}{3}}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Step 5.3.2

Cancel the common factor of $2$ .

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Step 5.3.2.1

Factor $2$ out of $2 π$ .

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

Step 5.3.2.2

Cancel the common factor.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Step 5.3.2.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 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.

$2 x = 2 π - \frac{2 π}{3}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify.

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Step 7.1.1

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

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

Step 7.1.2

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 7.1.4

Multiply $3$ by $2$ .

$2 x = \frac{6 π - 2 π}{3}$

Step 7.1.5

Subtract $2 π$ from $6 π$ .

$2 x = \frac{4 π}{3}$

Step 7.2

Divide each term in $2 x = \frac{4 π}{3}$ by $2$ and simplify.

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Step 7.2.1

Divide each term in $2 x = \frac{4 π}{3}$ by $2$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{4 π}{3}}{2}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{4 π}{3} \cdot \frac{1}{2}$

Step 7.2.3.2

Cancel the common factor of $2$ .

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Step 7.2.3.2.1

Factor $2$ out of $4 π$ .

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

Step 7.2.3.2.2

Cancel the common factor.

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

Step 7.2.3.2.3

Rewrite the expression.

$x = \frac{2 π}{3}$

Step 8

Find the period of $\cos (2 x)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 8.4.2

Divide $π$ by $1$ .

$π$

Step 9

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

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