Sun Local Hour Angle and Latitude

[moja_a]

Homework Statement

if:
H = Local Hour Angle
Lat = Latitude.
Dec = Sun Declination.

cos(H) = -sin(a)-sin(Lat)*sin(Dec) / cos(Lat)*cos(Dec)


I wand to get The value of Lat .

The Attempt at a Solution

I Tried to make it simple By :
1 - multiply both sides by the denominator
1- square both sides
2- Rewrite cos^2(L) as 1-sin^2(L)
3- Rearrange into a quadratic equation in sin(L).
And this is what I got:

Cos^2(H).Cos^2(D).Cos^2(L) = -Sin^2(a) - 2Sin(a).Sin(D).Sin(L) + Sin^2(L).Sin^2(D)

Cos^2(H).Cos^2(D).(1-Sin^2(L) )= -Sin^2(a) - 2Sin(a).Sin(D).Sin(L) + Sin^2(L).Sin^2(D)

Cos^2(H).Cos^2(D) - Cos^2(H).Cos^2(D).Sin^2(L) = - Sin^2(a) - 2Sin(a).Sin(D).Sin(L) + Sin^2(L).Sin^2(D)

Cos^2(H).Cos^2(D) = -Sin^2(a)- 2Sin(a).Sin(D).Sin(L) + Sin^2(L).Sin^2(D) + Cos^2(H).Cos^2(D).Sin^2(L)

Cos^2(H).Cos^2(D) = -Sin^2(a)- 2Sin(a).Sin(D).Sin(L) + Sin^2(L) . ( Sin^2(D) + Cos^2(H).Cos^2(D) )

then I am Stuck can you help me please.

 

[Redbelly98]

Welcome to Physics Forums.

So, is the problem asking you to solve the equation for Lat (or "L")?

As I read the equation, Lat appears in the form

sin(Lat) / cos(Lat)​

This can be replaced by a different, well-known trig function.

 

[moja_a]

Yes I used L = Lat
can you tell me about the function that replace it

 

[moja_a]

can you tell me about the function that replace it

 

[Redbelly98]

There are six standard, well-known trig functions. It is the one that is equivalent to

sin(θ) / cos(θ) = ___?​

Surely you know this?

 

[moja_a]

may be I wrote the formula incorrectly

Cos H = \frac{sin(a)-sin(L)*Sin(D)}{Cos(L)*Cos(D)}

I want to get the value of sin(L)

 

[gneill]

cos(H) = \frac{sin(a)-sin(L)*sin(D)}{cos(L)*cos(D)}

cos(L) = \frac{sin(a)-sin(L)*sin(D)}{cos(H)*cos(D)}

cos(L) = \frac{sin(a)}{cos(H)*cos(D)} \; - \; \frac{sin(D)}{cos(H)*cos(D)}sin(L)

define constants A and B appropriately and the equation becomes:

cos(L) = A - B sin(L)

You could square both sides at this point, change cos²(L) to 1 - sin²(L) and solve the quadratic for sin(L).

 

[RTW69]

Or Sin(L)=[Sin(a)-Cos(L)*Cos(H)*Cos(D)]/Sin(D)

 

[gneill]

Or Sin(L)=[Sin(a)-Cos(L)*Cos(H)*Cos(D)]/Sin(D)

How would that help? You've got a cos(L) on the RHS. You're trying to solve for L (eventually).

 

[RTW69]

You are correct, I should have read the problem statement a little closer.