Find the Roots (Zeros) y=2sin(4x)

$y = 2 \sin (4 x)$

Step 1

Set $2 \sin (4 x)$ equal to $0$ .

$2 \sin (4 x) = 0$

Step 2

Solve for $x$ .

Divide each term in $2 \sin (4 x) = 0$ by $2$ and simplify.

Divide each term in $2 \sin (4 x) = 0$ by $2$ .

$\frac{2 \sin (4 x)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sin (4 x)}{2} = \frac{0}{2}$

Divide $\sin (4 x)$ by $1$ .

$\sin (4 x) = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$\sin (4 x) = 0$

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

$4 x = \arcsin (0)$

Simplify the right side.

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

$4 x = 0$

Divide each term in $4 x = 0$ by $4$ and simplify.

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x = 0$

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.

$4 x = π - 0$

Solve for $x$ .

Simplify.

Multiply $- 1$ by $0$ .

$4 x = π + 0$

Add $π$ and $0$ .

$4 x = π$

Divide each term in $4 x = π$ by $4$ and simplify.

Divide each term in $4 x = π$ by $4$ .

$\frac{4 x}{4} = \frac{π}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4}$

Divide $x$ by $1$ .

$x = \frac{π}{4}$

Find the period of $\sin (4 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

The period of the $\sin (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

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

Consolidate the answers.

$x = \frac{π n}{4}$ , for any integer $n$

Step 3

Find the Roots (Zeros) y=2sin(4x)

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