Question

a ship leaves port r and sails to port t. bearing of s from r : 112 bearing of t from s : 033

Answer 1

I want to make sure I understand the problem before computing: you’ve given
- the bearing of S from R is 112° (i.e. the line R→S is 112° clockwise from North), and
- the bearing of T from S is 033° (i.e. the line S→T is 33° clockwise from North).

Do you want the bearing of T from R? If so: with only those two bearings (and no distances) the bearing of T from R is not uniquely determined — it depends on the lengths RS and ST. If you can give either RS and ST (or their ratio) I can give the exact bearing.

If you want the general method, here it is:

1. Put R at the origin. Represent each leg by its components (east = x, north = y).
- For a leg of length L at bearing β (degrees clockwise from north):
x = L · sin β
y = L · cos β

2. Let RS = a, ST = b. Then the coordinates of T relative to R are
x_RT = a·sin(112°) + b·sin(33°)
y_RT = a·cos(112°) + b·cos(33°)

3. The bearing of T from R is the angle clockwise from north to the vector (x_RT, y_RT):
bearing = arctan2(x_RT, y_RT) converted to degrees and normalized to 0–360°.
(Here arctan2(x,y) gives the angle whose tangent is x/y, taking quadrant into account.)

Example (if RS = ST, i.e. a = b = 1):
- x = sin112° + sin33° ≈ 1.4712
- y = cos112° + cos33° ≈ 0.4641
- bearing ≈ arctan2(1.4712, 0.4641) ≈ 72.5°
So in that case the bearing of T from R ≈ 073°.

Tell me the distances (or their ratio), or confirm you want the equal-length example, and I’ll compute the exact bearing.