study of sunspots. spots in a year min. is 10, max is 110. between max's of 1750 & 1948, 18 cycles occurred. Find a sinusoidal equation to model this.
My equation looks something like this so far:
y=50sin(2pi/11(x-d))+60
so How do I get the phase shift? d? I am confused about what would be x and y to plug into the formula and find d? Because the max occurs at 1750, but what would be the x and y values to sub in and why?
y=50sin( 2pi(x-d)/11 )+60
when x = 1750, you have a max and sin (2 pi(1750-d)/11) = 1
or 2 pi (1750-d)/11 = pi/2 (90 degrees for max of sin function)
1750-d =11/4 (in other words a quarter of a cycle after start)
d = 1750 - 11/4
How did you get 1750-d=11/4?
I mean, so would d = 1747.25
ie
y=50sin(2pi/11(x-1747.25))+60
To find the phase shift (d) in the sinusoidal equation, we need to determine the values of x and y to substitute into the equation. In this case, x represents the year, and y represents the number of sunspots observed in that particular year.
You mentioned that the maximum number of sunspots observed per year is 110 and the minimum is 10. Let's assume the maximum number of sunspots occurs at x = A, which is the peak year of each cycle.
Given that there were 18 cycles between the maximums of 1750 and 1948, we can calculate the average length of each cycle.
Average cycle length = (1948 - 1750) / 18 = 10.444 years per cycle (approximately)
Now, let's assign arbitrary x and y values to find the value of d:
1. Select one peak year, let's say 1750, and assume it occurs at x = 0.
- The corresponding y-value would be the maximum number of sunspots observed in that year, which is y = 110.
2. Select another peak year, let's say 1761 (since the average length of a cycle is 10.444 years).
- The corresponding x-value would be 1761 - 1750 = 11 years from the previous peak year.
- The y-value can be randomly chosen, but let's assign it y = 100 for this example.
Now we have two points: (0, 110) and (11, 100). We can substitute these values into the equation:
110 = 50sin(2π/11(0 - d)) + 60
100 = 50sin(2π/11(11 - d)) + 60
We can solve these two equations simultaneously to find the value of d. Once we find the value of d, we can substitute it back into the equation to obtain the complete sinusoidal equation.