figured out how to search the distance between two points on the map, but I can’t understand how to solve the opposite task.
There are three points on the map with known coordinates A (0,0) B (0.90) C (90.0)
and are known to the distance from them to point d, the question is how to find the coordinates of this point. As I understood, it is enough to solve the spherical triangle ABD, and the distance with use only to determine the pole. But with the calculation of this, problems arose.
UPD led to the task to solve the ABC triangle, following the example https://www.math10.com/ru/vysshaya-matematika/sfericheskii-treugolnik/sfericheskii-treugolnik.html
a (0,0) B (0.90) C (? ,?) AC = 8000; AB = 10007; Bc = 4000; a = 1.255; B = 1.571; C = 0.628; COS A = (COS A - COS B * COS C) / SIN B * SIN C (0.370-0 * 0.809) /1*.587=0.630
I can not find which corner meets this cosine, and how can you get bearing to calculate the coordinates? Or did I do something wrong?
Answer 1, Authority 100%
You know the angular lengths of the sides (distances divided by the land radius)
Apply spherical cosine theorem , To find the inner corner at a
cos a = cos b ⋅ cos c + sin b ⋅ sin c ⋅ cos a
from here (using D, and not with, because triangle ABD)
a = arccos ((COS A - COS B ⋅ COS D) / (SIN B ⋅ SIN D)) for ad = 8000; AB = 10007; Bd = 4000; opposing top and side a = bd / 6371 = 0.6278 B = AD / 6371 = 1.255 D = AB / 6371 = 1.571 A = ArcCOS ((COS A - COS B ⋅ COS D) / (SIN B ⋅ SIN D)) = ArcCOS ((0.8093 - 0.3106 ⋅ 0) / (0.9505 ⋅ 1)) = ArcCOS (0.851) = 0.552 radians = 31.68 degrees
This is azimuth from point A (in this case, azimuth is immediately, since A and B at the equator)
then use distance and azimuth (a) to determine the coordinates D, as shown Here in the section
Destination Point Given Distance and Bearing from Start Point
φ2 = asin (sin φ1 ⋅ cos δ + cos φ1 ⋅ sin δ ⋅ cos θ) λ2 = λ1 + atan2 (sin θ ⋅ sin δ ⋅ cos φ1, cos δ - sin φ1 ⋅ sin φ2) Where φ is Latitude, λ is Longitude, θ is the Bearing (Clockwise Fråm North), Δ is the angular distance d / r; D Being The Distance Travelled, R The Earth's Radius Lat = arcsin (sin φ1 ⋅ cos δ + cos φ1 ⋅ sin δ ⋅ cos θ) = Arcsin (0 * COS B + 1 * SIN B * SIN (0.552)) = Arcsin (0.9505 * 0.524) = Arcsin (0.4984) = 29.9 degrees (SS or Yush) And then for Lon
Selecting a hemisphere for D – Yes, as already said, away from the pole