Find All Complex Solutions cos(2x)+cos(x)=-1

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

Step 1

Add $1$ to both sides of the equation.

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

Step 2

Simplify the left side of the equation.

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Combine the opposite terms in $2 \cos^{2} (x) - 1 + \cos (x) + 1$ .

Add $- 1$ and $1$ .

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

Add $2 \cos^{2} (x)$ and $0$ .

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

Step 3

Factor $\cos (x)$ out of $2 \cos^{2} (x) + \cos (x)$ .

Factor $\cos (x)$ out of $2 \cos^{2} (x)$ .

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

Raise $\cos (x)$ to the power of $1$ .

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

Factor $\cos (x)$ out of $\cos^{1} (x)$ .

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

Factor $\cos (x)$ out of $\cos (x) (2 \cos (x)) + \cos (x) \cdot 1$ .

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

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

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

Step 5

Set $\cos (x)$ equal to $0$ and solve for $x$ .

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Solve $\cos (x) = 0$ for $x$ .

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

$x = \arccos (0)$

Simplify the right side.

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

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.

$x = 2 π - \frac{π}{2}$

Simplify $2 π - \frac{π}{2}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

Step 6

Set $2 \cos (x) + 1$ equal to $0$ and solve for $x$ .

Set $2 \cos (x) + 1$ equal to $0$ .

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

Solve $2 \cos (x) + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 \cos (x) = - 1$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $\cos (x)$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

Simplify the right side.

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

$x = \frac{2 π}{3}$

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.

$x = 2 π - \frac{2 π}{3}$

Simplify $2 π - \frac{2 π}{3}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Simplify the numerator.

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

Step 7

The final solution is all the values that make $\cos (x) (2 \cos (x) + 1) = 0$ true.

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

Step 8

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

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

Find All Complex Solutions cos(2x)+cos(x)=-1

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