Solve for θ sin(2theta)=0

$\sin (2 θ) = 0$

Step 1

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

$2 θ = \arcsin (0)$

Step 2

Simplify the right side.

The exact value of $\arcsin (0)$ is $0$ .

$2 θ = 0$

Step 3

Divide each term in $2 θ = 0$ by $2$ and simplify.

Divide each term in $2 θ = 0$ by $2$ .

$\frac{2 θ}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{0}{2}$

Divide $θ$ by $1$ .

$θ = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$θ = 0$

Step 4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$2 θ = π - 0$

Step 5

Solve for $θ$ .

Simplify.

Multiply $- 1$ by $0$ .

$2 θ = π + 0$

Add $π$ and $0$ .

$2 θ = π$

Divide each term in $2 θ = π$ by $2$ and simplify.

Divide each term in $2 θ = π$ by $2$ .

$\frac{2 θ}{2} = \frac{π}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{π}{2}$

Divide $θ$ by $1$ .

$θ = \frac{π}{2}$

Step 6

Find the period of $\sin (2 θ)$ .

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 $1$ .

$π$

Step 7

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

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

Step 8

Consolidate the answers.

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

Solve for θ sin(2theta)=0

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