Law of Sines and Law of Cosines help

[Kimisaishime]

Homework Statement

Ok the problem states Station A observes the fire at an angel of 52 and station B spots the fire at an angle of 41. The stations are 17 mi. apart. Find the distance along the line of sight from the fire to the station that is closest to the fire.

I'm confused I don't hink I have enough to do the Law of Cosines, but I also feel that I don't have enough to do the Law of Sines. I am so confused do I need to do something else. Like do I need to use the distance equation along with the the Law of Sines or the Law of Cosines. Or do I need to do something different entirely.

Homework Equations

The Attempt at a Solution

 

[symbolipoint]

Do you need to solve the problem in two different ways? I see direct possibility for Law Of Sines.

Your problem description conforms to a triangle, base points A and B. AB is 11 miles; Angle at B is 41 degrees; angle at A is 128 degrees (do you see why?); the angle opposite AB is 11 degrees (now, do you see why this too?)

You know all three angles and you know only one side (AB); you just want to know the the side that goes from the point opposite of AB to the point A.

 

[Architeuthis Dux]

Hi there,

Thank you for reaching out for help with your problem. It seems like you have a good understanding of the Law of Sines and Law of Cosines, but you're not sure which one to use in this situation. Let me try to break it down for you.

First, let's start by drawing a diagram to visualize the problem. We have Station A and Station B, which are 17 miles apart, and they are both observing a fire at different angles. The distance from the fire to each station is what we are trying to find. So, we can label the distance from the fire to Station A as x and the distance from the fire to Station B as y.

Next, let's think about what information we have and what we are trying to find. We know the angles at each station, but we don't know the side lengths. This means we can use the Law of Sines to solve for the side lengths. However, we also have the distance between the two stations, which we can use to set up a relationship between x and y. This is where we can use the Law of Cosines.

So, to summarize, we can use the Law of Sines to solve for x and y, and then we can use the Law of Cosines to relate x and y to the 17 miles between the two stations. This will give us enough information to solve for the distance along the line of sight from the fire to the station that is closest to the fire.

I hope this helps clarify things for you. Keep practicing and you'll become more comfortable with using these equations in different situations. Good luck with your homework!