Solve for θ cos(2theta)=0

$\cos (2 θ) = 0$

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

$2 θ = \arccos (0)$

Simplify the right side.

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

$2 θ = \frac{π}{2}$

Divide each term in $2 θ = \frac{π}{2}$ by $2$ and simplify.

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

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{2}$

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

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

$θ = \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$θ = \frac{π}{4}$

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.

$2 θ = 2 π - \frac{π}{2}$

Solve for $θ$ .

Simplify.

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

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

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$2 θ = \frac{2 π \cdot 2 - π}{2}$

Multiply $2$ by $2$ .

$2 θ = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$2 θ = \frac{3 π}{2}$

Divide each term in $2 θ = \frac{3 π}{2}$ by $2$ and simplify.

Divide each term in $2 θ = \frac{3 π}{2}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{3 π}{2}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{3 π}{2} \cdot \frac{1}{2}$

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$θ = \frac{3 π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$θ = \frac{3 π}{4}$

Find the period of $\cos (2 θ)$ .

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.

$\frac{2 π}{| 2 |}$

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 $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

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

Consolidate the answers.

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

Solve for θ cos(2theta)=0

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