Sinusodial Function: position P of sun on day d is P(d)=(sin((360/365)d-81) degrees

[MJFan]

Hi!
I am trying to solve this problem:

The position, P (d), of the Sun at sunset, in degrees north or south of due west, depends on the latitude and the day of the year, d. For a specific latitude, the position in terms of the day of the year can be modelled by the function:

. . . . .\(\displaystyle P(d)\, =\, 28\, \sin\left(\dfrac{360}{365}\, d\, -\, 81\right)^{\circ}\)

c) What is the equation of the axis of the curve, and what does it represent in this situation?

I have got this so far:

\(\displaystyle =\, \dfrac{28\, -\, 28}{2}\)

\(\displaystyle =\, 0\)

So I know that it's at y=0. However, I have no idea what it means.

If anyone could point me in the right direction that would be very helpful!

Thanks in advanced!

 

[ksdhart]

I'm not familiar with the term "axis of a curve." Can you please define what that means in the context of your class and/or your book? I fear I won't be of any help unless we're operating on the same page. If possible, please type out the definition word for word, to eliminate confusion. Thank you.

 

[MJFan]

I'm not familiar with the term "axis of a curve." Can you please define what that means in the context of your class and/or your book? I fear I won't be of any help unless we're operating on the same page. If possible, please type out the definition word for word, to eliminate confusion. Thank you.

My teacher want's us to call it that, and not the axis of symmetry. But as far as I can tell there the exact same thing. Anyway, it's the line that splits the function in half.

 

[stapel]

The position, P (d), of the Sun at sunset, in degrees north or south of due west, depends on the latitude and the day of the year, d. For a specific latitude, the position in terms of the day of the year can be modelled by the function:

. . . . .\(\displaystyle P(d)\, =\, 28\, \sin\left(\dfrac{360}{365}\, d\, -\, 81\right)^{\circ}\)

c) What is the equation of the axis of the curve, and what does it represent in this situation?

How does your book (or instructor) define the "axis" ("of the curve" or "of symmetry") for a sine wave?

I have got this so far:

\(\displaystyle =\, \dfrac{28\, -\, 28}{2}\)

What is the left-hand side, the thing to which this expression is equal? How did you get to this stage?

Please reply showing all of your work and reasoning (along with any other funky definitions or rules, possibly relevant to this exercise, with which your instructor is burdening you). Thank you!

 

[ksdhart]

Oh, okay. I suspected that might be the case, but I wanted to double check first. To begin finding the axis, I'd note the axis of the basic sine function. Sine is an odd function, so it's symmetrical about the origin (0,0), Such that sin(-x) = -sin(x). Does this symmetry hold for the modified sine function? In other words, is the following statement true?

\(\displaystyle 28sin\left(-\left[\frac{360x}{365}-\frac{81\pi }{180}\right]\right)=-28sin\left(\frac{360x}{365}-\frac{81\pi \:}{180}\right)\)

What does that tell you about the axis of symmetry? And how does that connect with what you found earlier?

 

[MJFan]

How does your book (or instructor) define the "axis" ("of the curve" or "of symmetry") for a sine wave?



What is the left-hand side, the thing to which this expression is equal? How did you get to this stage?

Please reply showing all of your work and reasoning (along with any other funky definitions or rules, possibly relevant to this exercise, with which your instructor is burdening you). Thank you!

So the (28-28)/2 comes from the equation she gave us to find the axis of symmetry. Which shows that the axis of symmetry equals 0. It's Gr. 11, so I don't know if I should mention that we're using degrees, and not radians.

To clarify the question a little more, I'm not sure what it means in terms of what does the equation of the curve represent in this question. Is it something like where the sun is neither up nor down? That's what I'm un-sure of.

 

[Ishuda]

So the (28-28)/2 comes from the equation she gave us to find the axis of symmetry. Which shows that the axis of symmetry equals 0. It's Gr. 11, so I don't know if I should mention that we're using degrees, and not radians.

To clarify the question a little more, I'm not sure what it means in terms of what does the equation of the curve represent in this question. Is it something like where the sun is neither up nor down? That's what I'm un-sure of.

I'm somewhat confused. I had always though that a (vertical) axis of symmetry was a line x = c such that
f(c-x) = f(c+x); x\(\displaystyle \ge\)0.
If this is true, then it can easily be shown that the axis of symmetry is not zero for P(d). However it is true that zero is an axis of 'anti-symmetry', i.e.
P(0-x)=-P(0+x)
where
x = \(\displaystyle (\frac{360}{365}\, d\, -\, 81)^{\circ}\)
but I think you want
P(c-d)=-P(c+d)
Also note that there are many axes of anti-symmetry.

EDIT: If you are having a hard time, I would suggest you plot P(d) and see if that gives you any ideas.

 

[MJFan]

I'm somewhat confused. I had always though that a (vertical) axis of symmetry was a line x = c such that
f(c-x) = f(c+x); x\(\displaystyle \ge\)0.
If this is true, then it can easily be shown that the axis of symmetry is not zero for P(d). However it is true that zero is an axis of 'anti-symmetry', i.e.
P(0-x)=-P(0+x)
where
x = \(\displaystyle (\frac{360}{365}\, d\, -\, 81)^{\circ}\)
but I think you want
P(c-d)=-P(c+d)
Also note that there are many axes of anti-symmetry.

EDIT: If you are having a hard time, I would suggest you plot P(d) and see if that gives you any ideas.

I ended up figuring it out. It was representing when the sun was due west.