Right Triangle Math question?

Question

a ship sailing in the direction S42 degress passes a point A directly east of a light house. If the angle of elevation from A to the top of the lighthouse is 19 degrees, find the angle of the elevation when the ship is closest to the light house.

Answer 1

h/d0 = tan19°
d0 = h/tan19°
d1 = d0sin42°
d1 = hsin42°/tan19°
h/d1 = tanθ
tanθ = htan19°/(hsin42°)
θ = arctan(tan19°/sin42°)
θ = arctan(0.34433/0.39245)
θ = 27.230°

Answer 2

i am not sure what direction S42 really means , i take it as 42 degrees from north towards eastern direction (may be i am wrong here but you can modify your calculation accordingly) that is approx, north-east direction.
at A angle of elevation is 19 degrees, if the light house is X metrs east of ship and take the top of the light house as point c and base of the light house as point B , ABC is a rectangular triangle hypotenuses AC makes 19 degree angle, AB= X meters
Therefore BC height of lighthouse= Xtan(19)

since the ship travels in the direction 42 degres from north AB makes an angle = 90 - 42 degrees = 48 degrees (this is easy because eastern and n-s directions are perpendicular)

ship is closest to light house at say a point D, at such a point BD should be perpendicular to the direction of ships travel, (shortest distance from a point to a line is perpendicular to the line)

now ABD is a rectangular triangle AB is the hypotenuses = x
angle DAB= 48 deg
so BD, the closest distance to the base of lighthouse is = x.sin(48)

THe elevation at this point could be found using triangle DBC also rectangular
The elevation is angle BDC TAN(angle BDC)=BC/BD=xtan19/xsin71=tan 19/sin48=.4633

angle(BDC)=tan^(-1)(.3642) = 24.9 deg