# Finding the center by area of a range of a sine wave

• Puggley
In summary, the person is seeking help with a math problem involving finding a point on a sine wave that divides the area between two other points on the wave equally. The solution involves using integrals and trigonometry to find the appropriate x value.
Puggley
hey.

I used to be quite talented at math, but I've let my talent deteriorate in the ~10 years since my last calculus class.

I have a problem which I'm sure I would have devoured easily back in my school days, but am having trouble with now.

I am considering two points on a normal (0 to 2pi, amplitude of 1) sine wave, p1 and p2. What I want to do is select p3 in between them so that the area of the region between p1 and p3 is equal to the area of the region between p3 and p2.

THANX
Puggley

Hi Puggley!

You'll need to find the t that solves

$$\int_{p_1}^t{\sin(x)dx}=\int_t^{p_2}{\sin(x)dx}$$

(Note: I'm calculating the orientated area here, that is: point below the x-axis count as negative area. If you don't want this, you'll need to take the absolute value).

Calculating the integrals gives us

$$[-\cos(x)]_{p_1}^t=[-\cos(x)]_t^{p_2}$$

And thus, you need to find x such that

$$\cos(p_1)-\cos(t)=\cos(t)-\cos(p_2)$$

And thus

$$\cos(t)=\frac{\cos(p_1)+\cos(p_2)}{2}$$

Hence, we take

$$t=arccos\left(\frac{\cos(p_1)+\cos(p_2)}{2}\right)$$

I'd like to add the second possible solution $2\pi - t$ and choose the one that is between p1 and p2.

## 1. What is the center of a sine wave?

The center of a sine wave refers to the midline of the wave, which is the horizontal line that runs through the center of the wave. It represents the average value of the wave and is used as a reference point for measuring the amplitude of the wave.

## 2. How do you find the center of a sine wave by area?

The center of a sine wave can be found by calculating the area under the curve of the wave and then dividing it by the total width of the wave. This will give you the distance from the midline to the center of the wave.

## 3. Why is finding the center of a sine wave important?

Finding the center of a sine wave is important because it helps us understand the properties of the wave, such as its amplitude and frequency. It also allows us to accurately plot and analyze the wave, which has numerous applications in fields such as physics, engineering, and mathematics.

## 4. Can the center of a sine wave change?

Yes, the center of a sine wave can change depending on the amplitude and frequency of the wave. If the amplitude increases, the center of the wave will shift upwards, and if the frequency changes, the center of the wave will shift horizontally.

## 5. How is the center of a sine wave related to its amplitude?

The center of a sine wave is directly related to its amplitude. As the amplitude increases, the distance between the center of the wave and the midline increases, and vice versa. This relationship is important in understanding the characteristics and behavior of the sine wave.

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