The Laws of Sines Cosines Determine how far a boat is from shore

Question

To determine how far a boat is from shore, two radar stations have been erected 1 mile apart on the shoreline. These stations then measure the angles out to the boat as shown in Figure 2. Find the (perpendicular) distance of the boat from shore.

Answer 1

Let's say that the angles given are the math angles and that the shore is the x- axis. θ_L and 180-θ_R are the angles at the base of the triangle that is formed from the two stations (Left and Right) and the boat. You could find the third angle (at the boat) by 180-(180-θ_R+θ_L) = θ_R-θ_L = θ_Boat

You can use sine law to find all the sides: sinθ_B/1 = sinθ_L/d_R = sin(180-θ_R)/d_L where d_i is the distance from the right station(R) and left station(L). Note that the sine law is for the side opposite the angle.

Once you know one of the sides (let's say the left) : height of triangle = d_Lsinθ_L