Solve for x cos(3x)+cos(x)=0

$\cos (3 x) + \cos (x) = 0$

Step 1

Simplify the left side of the equation.

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

$4 \cos^{3} (x) - 3 \cos (x) + \cos (x) = 0$

Add $- 3 \cos (x)$ and $\cos (x)$ .

$4 \cos^{3} (x) - 2 \cos (x) = 0$

Step 2

Factor $2 \cos (x)$ out of $4 \cos^{3} (x) - 2 \cos (x)$ .

Factor $2 \cos (x)$ out of $4 \cos^{3} (x)$ .

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

Factor $2 \cos (x)$ out of $- 2 \cos (x)$ .

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

Factor $2 \cos (x)$ out of $2 \cos (x) (2 \cos^{2} (x)) + 2 \cos (x) \cdot - 1$ .

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

Step 3

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

$\cos (x) = 0$

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

Step 4

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 5

Set $2 \cos^{2} (x) - 1$ equal to $0$ and solve for $x$ .

Set $2 \cos^{2} (x) - 1$ equal to $0$ .

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

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

Add $1$ to both sides of the equation.

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

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

Divide each term in $2 \cos^{2} (x) = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

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

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

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

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

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the right side.

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

$x = \frac{π}{4}$

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.

$x = 2 π - \frac{π}{4}$

Simplify $2 π - \frac{π}{4}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Simplify the numerator.

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

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.

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

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

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

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

Simplify the right side.

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

$x = \frac{3 π}{4}$

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.

$x = 2 π - \frac{3 π}{4}$

Simplify $2 π - \frac{3 π}{4}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Simplify the numerator.

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

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.

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

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 $n$

Consolidate the answers.

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

Step 6

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

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

Step 7

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

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

Solve for x cos(3x)+cos(x)=0

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