Solve for x cos(2x)=( square root of 2)/2

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

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.

Tap for more steps...

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

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

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{π}{4} \cdot \frac{1}{2}$

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

Tap for more steps...

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

$x = \frac{π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

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.

$2 x = 2 π - \frac{π}{4}$

Simplify the expression to find the second solution.

Tap for more steps...

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 $1$ .

Tap for more steps...

Combine.

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

Multiply $4$ by $1$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

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

Combine the 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.

Tap for more steps...

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

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

Cancel the common factor of $2$ .

Tap for more steps...

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 $\frac{7 π}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

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

$x = \frac{7 π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{7 π}{8}$

Find the period.

Tap for more steps...

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 2 |}$

Solve the equation.

Tap for more steps...

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 $2$ .

Tap for more steps...

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 both directions.

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

Do you need help with solving Solve for x cos(2x)=( square root of 2)/2? We can help you. You can write to our math experts in our application. The best solution for you is above on this page.

Evaluate ( log of 16)/( log of 4)

Find the Domain f(x)=-3

Find All Possible Roots/Zeros Using the Rational Roots Test x^3-4x^2-7x+10

Solve for x sin(5x)=0

Find All Possible Roots/Zeros Using the Rational Roots Test cos(2x)

Solve for x cos(2x)=( square root of 2)/2

100% Correct Answer from a Real Math Expert