Two lookout stations, which are 25 miles apart along the coast on a north-south line, spot an approaching yacht. One lookout station measures the direction to the yacht at N 33degrees E, and the other station measures the direction of the yacht at S 62degress E. How far is the yacht from each lookout station? How far is the yacht from the coast?

Draw a diagram. If the northern station is at A, and the southern is at B, and the yacht is at C,

angle C is 180-62-33 = 85°

Using the law of sines,

a/sin62° = b/sin33° = 25/sin85°
a (distance from B) = 22.16
b (distance from A) = 13.67
distance from coast is a sin33° = b sin62° = 12.07

To find the distances, we can use trigonometry. Let's start by finding the distance between the yacht and each lookout station.

First, let's label the lookout station where the direction measured is N 33 degrees E as station A, and the station with the direction measured as S 62 degrees E as station B.

To find the distance between the yacht and station A, we can use the concept of right triangles. Draw a line connecting the yacht with station A, and label the distance as x.

Now, let's analyze the direction measurements. The direction from station A to the yacht is given as N 33 degrees E. This means that the angle between the line connecting the yacht and station A and the north direction is 33 degrees. We can represent this as angle NAC (let AC be the line connecting the yacht and station A).

Similarly, the direction from station B to the yacht is given as S 62 degrees E. This means that the angle between the line connecting the yacht and station B and the south direction is 62 degrees. We can represent this as angle SBC (let BC be the line connecting the yacht and station B).

Since the two lookout stations are 25 miles apart, we can consider AB as the base of a triangle, with an angle of 180 - (33 + 62) degrees at point B. This is because the angles in any triangle sum up to 180 degrees.

We can use the trigonometric function tangent (tan) to find the value of x. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.

In triangle ABC, tan(angle SBC) = x / AB. Rearranging this equation, x = AB * tan(angle SBC).

In triangle ABC, tan(angle NAC) = x / AB. Rearranging this equation, x = AB * tan(angle NAC).

Now let's calculate the values:

x for station A = AB * tan(angle NAC)
x for station B = AB * tan(angle SBC)

To find AB, we can use the Pythagorean theorem. AB^2 = AC^2 + BC^2.

In triangle ABC, AC = x + distance from station A to the coast, and BC = distance from station B to the coast.

Using the Pythagorean theorem:
AB^2 = (x + distance from station A to the coast)^2 + (distance from station B to the coast)^2

Now, with these equations, we can calculate the distances between the yacht and each lookout station, as well as the distance between the yacht and the coast.

Please provide the values for the distance from station A to the coast and the distance from station B to the coast.