Solve for x cos(two x) equals ( square root of two) First, divide two

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

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

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

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

$2 x = \frac{π}{4}$

Divide each term by $2$ and simplify.

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

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

The common factor should be canceled of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{π}{4} \cdot \frac{1}{2}$

First, multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

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Multiply $\frac{π}{4}$ and $\frac{1}{2}$ .

$x = \frac{π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

The 1st and 4th quadrants of the cosine function are positive. To calculate the second answer, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 x = 2 π - \frac{π}{4}$

Clarify the expression To calculate the second answer.

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

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

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

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Add.

$\frac{2 π \cdot 4}{1 \cdot 4} - \frac{π}{4}$

Multiply $4$ by $1$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

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

Add numerators over the common denominator.

$2 x = \frac{2 π \cdot 4 - π}{4}$

Multiply $4$ by $2$ .

$2 x = \frac{8 π - π}{4}$

Subtract $π$ from $8 π$ .

$2 x = \frac{7 π}{4}$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = \frac{7 π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{7 π}{4} \cdot \frac{1}{2}$

The common factor $2$ should be canceled.

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Cancel the common factor.

$\frac{2 x}{2} = \frac{7 π}{4} \cdot \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{7 π}{4} \cdot \frac{1}{2}$

Multiply $\frac{7 π}{4} \cdot \frac{1}{2}$ .

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Multiply $\frac{7 π}{4}$ and $\frac{1}{2}$ .

$x = \frac{7 π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{7 π}{8}$

Calculate the period.

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Evaluate the period of function using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 2 |}$

Find solution to the equation.

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

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

$x = \frac{π}{8} + π n, \frac{7 π}{8} + π n$ , for any integer $n$

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