Math Is Fun Forum

Thank you, Bob. You gave me homework to think of

Yes, Thank you. It inspired me to think of this project (tracking the sun horizontally).

You are right. It is not simple as we like it to be.
I hope the formulas (the set of formulas) will give the azimuth and altitude (maximum, I guess) in function of date and time only, besides the laltitude and longitude of the observer

I guess, the horizontal limits will be made around ±60° for practical reasons (mechanical).

A + iB could be seen as the sum of two vectors on two perpendicular axes.
A on the real axis (horizontal).
B on the imaginary axis (vertical).
The modulus = the length of the sum vector = sqrt(A^2 + B^2) = r
The argument = its angle with the horizontal axis = arctan(B/A) = θ

The new form is usually written as:
r*e^iθ

You did help a lot. Thank you, Bob.

Therefore, the declination of the sun is now:
δ = 23.45° * sin( (360° / 365) * (n + 284) )
where:
n= 31+28+2 = 61 days

δ = 23.45° * sin( (360° / 365) * (61 + 284) ) = -7.914911995°

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the Sun's maximum altitude at Aleppo city is:
A=90−(φ−δ)
where:
φ = the latitude of the observer (degrees) = 36° 12' 7'' = 36.20194444°
δ = -7.914911995°

A= 90 - (36.20194444 + 7.914911995) ≈ 45.9°

Here, it is around 5.30 PM. So, tomorrow noon, I will try to measure the angle A (though for n=62, A≈46.3°)

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As I mentioned earlier, to simplify the positioner installation, I will try to move automatically the solar panel, horizontally only (The vertical movement will have to be done manually, every 3 months, for example).
In this case, if the controller's clock is set to 12 noon when the sun is at mid sky (at its maximum altitude), is it possible to calculate the sun deviation angle, θ, at 1 PM? Edited: Sorry again, you gave me the answer already which is 15° / hour (=360/24)

I guess this is the solution:

Appetizer                           
H1:    01    02    03    04    05    06    07
H2:    08    09    10    11    12    13    14
H3:    15    16    17    18    19    20    21
H4:    22    23    24    25    26    27    28
                           
Soup                           
H1:    15    23    24    25    26    27    28
H2:    22    16    17    18    19    20    21
H3:    01    09    10    11    12    13    14
H4:    08    02    03    04    05    06    07
                           
Main                           
H1:    22    09    10    11    12    13    14
H2:    15    02    03    04    05    06    07
H3:    08    23    24    25    26    27    28
H4:    01    16    17    18    19    20    21
                           
Dessert                           
H1:    08    16    17    18    19    20    21
H2:    01    23    24    25    26    27    28
H3:    22    02    03    04    05    06    07
H4:    15    09    10    11    12    13    14

Isn’t it a solution?

Appetizer                           
H1:    01    02    03    04    05    06    07
H2:    08    09    10    11    12    13    14
H3:    15    16    17    18    19    20    21
H4:    22    23    24    25    26    27    28
                           
Soup                           
H1:    01    09    10    11    12    13    14
H2:    08    02    03    04    05    06    07
H3:    15    23    24    25    26    27    28
H4:    22    16    17    18    19    20    21
                           
Main                           
H1:    01    16    17    18    19    20    21
H2:    08    23    24    25    26    27    28
H3:    15    02    03    04    05    06    07
H4:    22    09    10    11    12    13    14
                           
Dessert                           
H1:    01    23    24    25    26    27    28
H2:    08    16    17    18    19    20    21
H3:    15    09    10    11    12    13    14
H4:    22    02    03    04    05    06    07

We used to hear: "Necessity is the mother of invention/innovation"
This means that an invention or innovation is expected to be useful to the person who needed it in the first place.

In fact, this saying applies on my work.
Once a while, I have to think out of the box when I need to design something that no one else seemed to have an interest about it.
And when I try to present what I did, no one, as expected, has an interest to know/understand it.

Similarly, if one presents his invention or innovation that I don't need, I cannot focus on it while I try to solve what I need.

This is how life is since always.

Hi Bob,

Your solution is better since it doesn't use trig identities. Thank you.

Kerim

I believe you were able to get it, if you had enough free time.

The general form of the OP equation:
tan(A) = 2*tan(2*A-90) - tan(90-A)  {equation 0}

Let us start with the identity:
tan(a-b) = [ tan(a) - tan(b) ] / [ 1 + tan(a)*tan(b) ]

By replacing:
a = A
b = 90-A
We get:
tan[A-(90-A)] = [ tan(A) - tan(90-A) ] / [ 1 + tan(A)*tan(90-A) ]  {equation 1}

Now let us evaluate the product: tan(A)*tan(90-A)
By using the identity:
tan(a+b) = [ tan(a) + tan(b) ] / [ 1 - tan(a)*tan(b) ]

We get:
tan[A+(90-A)] = [ tan(A) + tan(90-A) ] / [ 1 - tan(A)*tan(90-A) ]
tan(90) = [ tan(A) + tan(90-A) ] / [ 1 - tan(A)*tan(90-A) ]

But tan(90) = ∞
∞ = [ tan(A) + tan(90-A) ] / [ 1 - tan(A)*tan(90-A) ] which means that the denominator of the fraction must be zero. {x/0 = ∞}
1 - tan(A)*tan(90-A) = 0
tan(A)*tan(90-A) = 1

After replacing tan(A)*tan(90-A) by 1 in {equation 1}, we get:
tan[A-(90-A)] = [ tan(A) - tan(90-A) ] / ( 1 + 1 )
tan[A-(90-A)] = [ tan(A) - tan(90-A) ] / 2
2*tan(2*A-90) = tan(A) - tan(90-A)

tan(A) = 2*tan(2*A-90) + tan(90-A) which is similar to {equation 0}

I am a bit busy now so I can just give a hint:
Try to draw the distance travelled by a falling object (a=g=10m/s2) versus time (assume that the downwards direction of the movement is positive). I don't think the graph will be a straight line.

Kerim