2. two look out situations, which are 25 miles apart along the coast on a north-south shoreline, spot an approaching yacht. One lookout station measures the direction to the yacht at N33 degrees E, and the other station measures the direction of the yacht at S62 degrees E. How far is the yacht from each lookout station? how far is the yacht from the coast?

make a sketch, label the side across from the 33º angle as x, and the side across the 62º as y.

Using the sine law
x/sin33 = 25/sin85

do y the same way.

draw a perpendicular from the boat to shore, giving you a right-angled triangle, call it h

sin 62 = h/x
h = xsin62, and since we know x ....

To solve this problem, we can use the concept of trigonometry and the properties of triangles. Let's break the problem down step by step:

First, let's draw a diagram to visualize the situation. We have two lookout stations, labeled A and B, which are 25 miles apart along the coast. The yacht is represented by the point Y, and we want to find the distances between the yacht and each lookout station, as well as the distance between the yacht and the coast.

```
Y
|
|
A-------------25 miles-------------B
```

Now, let's look at the given information:
- Lookout station A measures the direction to the yacht at N33 degrees E.
- Lookout station B measures the direction of the yacht at S62 degrees E.

To find the distance between the yacht and each lookout station, we can use the concept of trigonometry. Specifically, we will use the tangent function.

Let's consider lookout station A first. The measured direction is N33 degrees E. This means that the angle between the line connecting the yacht and the lookout station and the north direction is 33 degrees. Let's call this angle theta.

Using trigonometry, we can write the following equation:
tan(theta) = (distance between A and Y) / (distance between A and B)

Since the distance between A and B is given as 25 miles, we can rewrite the equation:
tan(theta) = (distance between A and Y) / 25

Similarly, for lookout station B, the measured direction is S62 degrees E. The angle between the line connecting the yacht and the lookout station and the south direction is 62 degrees. Let's call this angle phi.

Using trigonometry, we can write the following equation:
tan(phi) = (distance between B and Y) / (distance between A and B)

Again, using the given distance between A and B (25 miles), we can rewrite the equation:
tan(phi) = (distance between B and Y) / 25

Now, we have two equations with two unknowns (distance between A and Y and distance between B and Y). We can solve this system of equations to find the distances.

To do this, isolate the distance between A and Y in the first equation:
(distance between A and Y) = 25 * tan(theta)

And isolate the distance between B and Y in the second equation:
(distance between B and Y) = 25 * tan(phi)

Finally, to find the distance between the yacht and the coast, we will use the law of sines. In the triangle formed by the yacht, lookout station A, and the coast, we have the following information:
- The measure of angle at the yacht is 180 - (theta + 90) degrees.
- The distance between A and Y is given by 25 * tan(theta).

Using the law of sines, we can write the following equation:
(distance between A and Y) / sin(angle at yacht) = (distance between Y and coast) / sin(90 degrees)

Since sin(90 degrees) is equal to 1, we can simplify the equation to:
(distance between A and Y) = distance between Y and coast

Therefore, we can conclude that the distance between the yacht and the coast is equal to the distance between A and Y, which is 25 * tan(theta).

To summarize:
- The distance between the yacht and lookout station A is 25 * tan(theta).
- The distance between the yacht and lookout station B is 25 * tan(phi).
- The distance between the yacht and the coast is 25 * tan(theta).

Please note that in order to get the final numerical values, you need to substitute the values of theta and phi into the equations and perform the calculations.