Solve for ? square root of 2cos(x)^2+cos(x)=0

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Trigonometry

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

Factor the left side of the equation.

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

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

$\cos (x) (\sqrt{2} \cos (x)) + \cos (x)$

Multiply by $1$ .

$\cos (x) (\sqrt{2} \cos (x)) + \cos (x) \cdot 1$

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

$\cos (x) (\sqrt{2} \cos (x) + 1)$

Replace the left side with the factored expression.

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

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

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

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

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

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

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

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

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

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

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

$\cos (x) = 0$

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

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$\cos (x) = 0$

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

$x = \arccos (0)$

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 $2 π - \frac{π}{2}$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{2 π}{1} \cdot \frac{2}{2} - \frac{π}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{2 π \cdot 2}{1 \cdot 2} - \frac{π}{2}$

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Find the period.

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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.

$\frac{2 π}{| 1 |}$

Solve the equation.

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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 $n$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

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

Subtract $1$ from both sides of the equation.

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

Divide each term by $\sqrt{2}$ and simplify.

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

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

Cancel the common factor of $\sqrt{2}$ .

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Cancel the common factor.

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

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

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

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

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

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

Combine and simplify the denominator.

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Multiply $\frac{- 1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

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

Move the negative in front of the fraction.

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

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

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

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 $2 π - \frac{3 π}{4}$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{2 π}{1} \cdot \frac{4}{4} - \frac{3 π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{2 π \cdot 4}{1 \cdot 4} - \frac{3 π}{4}$

Multiply $4$ by $1$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Simplify the numerator.

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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.

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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.

$\frac{2 π}{| 1 |}$

Solve the equation.

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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 $n$

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

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

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

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

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