Explanation:
the law of sine
a/sin(A) = b/sin(B) = c/sin(C)
where the sides and associated angles are always opposite to each other.
and the sum of all angles in a triangle are always 180°.
sin(90) = 1.
4.
there are multiple solutions, as the direction they are going has no relevance to their position to each other right now.
and the angles of depression can be forward or backwards of the plane.
the larger the angle of depression, the closer the ship is.
for the closer ship we get
height/sin(35) = ground distance / sin(180-90-35)
265/sin(35) = ground distance / sin(55)
ground distance (plane to closer ship) =
= 265×sin(55)/sin(35) = 378.4592218... m
for the farther ship we get
height/sin(25) = ground distance / sin(180-90-25)
265/sin(25) = ground distance / sin(65)
ground distance (plane to closer ship) =
= 265×sin(65)/sin(25) = 568.2943339... m
the distance of both ships is
568.2943339... - 378.4592218... = 189.8351121... m
now, this distance applies to the case, where both ships are on the same side of the plane.
if they are on different sides of the plane, then we need to add their individual distances from the plane to get the distance of both ships from each other :
568.2943339... + 378.4592218... = 946.7535557... m
5.
another right-angled triangle.
the string is the Hypotenuse (baseline).
the height and the ground distance are the legs.
so, we get
100/sin(90) = height/sin(52) = 100
= ground distance / sin(180-90-52)
height = 100×sin(52) = 78.80107536... m
ground distance = 100×sin(38) = 61.56614753... m