Find the Roots (Zeros) y=2sin(x-pi/2)

$y = 2 \sin (x - \frac{π}{2})$

Step 1

Set $2 \sin (x - \frac{π}{2})$ equal to $0$ .

$2 \sin (x - \frac{π}{2}) = 0$

Step 2

Solve for $x$ .

Divide each term in $2 \sin (x - \frac{π}{2}) = 0$ by $2$ and simplify.

Divide each term in $2 \sin (x - \frac{π}{2}) = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $\sin (x - \frac{π}{2})$ by $1$ .

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

Simplify the right side.

Divide $0$ by $2$ .

$\sin (x - \frac{π}{2}) = 0$

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

$x - \frac{π}{2} = \arcsin (0)$

Simplify the right side.

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

$x - \frac{π}{2} = 0$

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2}$

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.

$x - \frac{π}{2} = π - 0$

Solve for $x$ .

Subtract $0$ from $π$ .

$x - \frac{π}{2} = π$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{π}{2}$ to both sides of the equation.

$x = π + \frac{π}{2}$

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

$x = π \cdot \frac{2}{2} + \frac{π}{2}$

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} + \frac{π}{2}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Find the period of $\sin (x - \frac{π}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Consolidate the answers.

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

Step 3

Find the Roots (Zeros) y=2sin(x-pi/2)

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