# Thread: Find angle of latitude: distance between P (63° N, 70°E), R (63° N, x° E) is 900km

1. ## Find angle of latitude: distance between P (63° N, 70°E), R (63° N, x° E) is 900km

Hi I've done a question on latitude and longitude.
It states : Take the radius of the earth to be 6400km and pi to be 3.142.
(i)The distance between Town P (63° N, 70°E) and Town R (63° N, x° E) is 900km.
If Town R is east of Town P, determine x°E, the longitude of R, to one decimal place.

Now I've done this using this formula: let (c) be the circumference of the parallel of latitude, and (C) be the circumference of the earth.

So I know that c = C*cos(theta)
C is = 40217.6km

And the length of the arc of latitude QY is : l=c*(a/360)
so : 900km=40217.6km *cos63° *a/360
a= (900*360)/(40217.6*cos63°)
a= 17.7

So I would have thought that now all I had to do was subtract this from the angle of longitude I have 70° but apparently its when I add 17.7° to 70° I get x = 87.7° which is the answer, this I don't understand as all the other examples in my book subtract.

Now for the second part of the question, this I'm stuck like Ive never been before.
''The shortest distance between Town P (63° N,70° E) and Town Q (y° N, 110° W) is 6702.9km

Determine y° N, the latitude of Q. (The answer is 57° if this helps anyone) I have no clue how to do this.

Here is a picture attached of how I got part (i)l.jpg
Some useful formulas maybe,
''If you have two places on the same latitude of 24° N, and the angle subtended between them is 120°,(circumference of earth is 40000km)
The shortest distance can be found as follows.
First find circumference of the parallel of latitude which is c=C*cos(theta)
So: c=*cos24° *40000km =36560km.

SO length of the arc PQ is : l=c*(a/360)
so: l=36560*120/360= 12186.7 km to 1DP.''

Hopefully this example and picture give some insight.

That is just an example. For part (ii) there is a similar way to calculate the latitude, but my book shows only examples where the longitude is the same... and in this case both the latitude and longitude are different.

2. Can anyone help?

3. Originally Posted by richiesmasher
Hi I've done a question on latitude and longitude.
It states : Take the radius of the earth to be 6400km and pi to be 3.142.
(i)The distance between Town P (63° N, 70°E) and Town R (63° N, x° E) is 900km.
If Town R is east of Town P, determine x°E, the longitude of R, to one decimal place.
...
a= 17.7

So I would have thought that now all I had to do was subtract this from the angle of longitude I have 70° but apparently its when I add 17.7° to 70° I get x = 87.7° which is the answer, this I don't understand as all the other examples in my book subtract.
I agree with 17.7°. But R is east of P, and P's longitude is 70° east, so why wouldn't you add 17.7 to 70? It's just like a number line: If P is at +70, and R is 17.7 to the right of that, then R is at 87.7.

Originally Posted by richiesmasher
Now for the second part of the question, this I'm stuck like Ive never been before.
''The shortest distance between Town P (63° N,70° E) and Town Q (y° N, 110° W) is 6702.9km

Determine y° N, the latitude of Q. (The answer is 57° if this helps anyone) I have no clue how to do this.

Here is a picture attached of how I got part (i)
Some useful formulas maybe,
''If you have two places on the same latitude of 24° N, and the angle subtended between them is 120°,(circumference of earth is 40000km)
The shortest distance can be found as follows.
First find circumference of the parallel of latitude which is c=C*cos(theta)
So: c=*cos24° *40000km =36560km.

SO length of the arc PQ is : l=c*(a/360)
so: l=36560*120/360= 12186.7 km to 1DP.''

Hopefully this example and picture give some insight.

That is just an example. For part (ii) there is a similar way to calculate the latitude, but my book shows only examples where the longitude is the same... and in this case both the latitude and longitude are different.
The example you quote seems irrelevant; it is about points with the same latitude.

Are there no examples in the book dealing with points for which neither latitude nor longitude is the same? I could invent a way to do this, or look up a formula, but presumably there is something you have learned that you are expected to use. One approach I might use, for example, would be to find the angle between the position vectors of the points.

4. Originally Posted by Dr.Peterson
I agree with 17.7°. But R is east of P, and P's longitude is 70° east, so why wouldn't you add 17.7 to 70? It's just like a number line: If P is at +70, and R is 17.7 to the right of that, then R is at 87.7.

The example you quote seems irrelevant; it is about points with the same latitude.

Are there no examples in the book dealing with points for which neither latitude nor longitude is the same? I could invent a way to do this, or look up a formula, but presumably there is something you have learned that you are expected to use. One approach I might use, for example, would be to find the angle between the position vectors of the points.
To be honest I see no examples of where neither latitude nor longitude is the same.
I'm really stumped and this topic is so foreign to me...
I see things about finding the sector angle when the latitude is the same, by adding the two latitudes and subtracting from 180 degrees. I also see things about two different longitudes on the same meridian circle..

5. Originally Posted by richiesmasher
To be honest I see no examples of where neither latitude nor longitude is the same.
I'm really stumped and this topic is so foreign to me...
I see things about finding the sector angle when the latitude is the same, by adding the two latitudes and subtracting from 180 degrees. I also see things about two different longitudes on the same meridian circle..
This is a much harder problem than those others. As I said, I could suggest several different ways to do it, but I doubt you are ready for many of them.

Please tell us as much as you can about the context. Assuming these questions are in a chapter, what is its title, and what topics does it cover? If I were helping you in person, I would have grabbed your book some time ago!

More broadly, where are you in learning about trig? Have you learned anything about vectors, or spherical coordinates, or anything else that might be useful? At this point, we're rummaging around in your toolkit, looking for something you can use so we don't have to go out and buy something you don't have (that is, teach you something you haven't learned yet).

I also have to ask, since I don't recall what you've said before (and might confuse you with someone else), are you in a course carefully working through this book, or did you jump into a place that you might not be ready for?

6. Originally Posted by Dr.Peterson
This is a much harder problem than those others. As I said, I could suggest several different ways to do it, but I doubt you are ready for many of them.

Please tell us as much as you can about the context. Assuming these questions are in a chapter, what is its title, and what topics does it cover? If I were helping you in person, I would have grabbed your book some time ago!

More broadly, where are you in learning about trig? Have you learned anything about vectors, or spherical coordinates, or anything else that might be useful? At this point, we're rummaging around in your toolkit, looking for something you can use so we don't have to go out and buy something you don't have (that is, teach you something you haven't learned yet).

I also have to ask, since I don't recall what you've said before (and might confuse you with someone else), are you in a course carefully working through this book, or did you jump into a place that you might not be ready for?
The chapter is geometry 3, it starts off with circle theorems, then goes to trigonometrical identities, then goes to angles of elevation and depression, then goes to airplane course and track, then goes to latitude and longitude.
I'm self taught, no teacher, but I'm writing exams in May. This question is from a practice exam, so every question I come across I have to go and study and learn about, as I havent been to school for many years, and can't remember everything, but I've come a good bit into this, and this is a topic i don't even remember from school, it's usually in Section 2 of the exam that I have to write, of which has 6 questions and I only have to do two.

So definitely it's an optional topic because all of my friends who write this exam don't even recall it, like they recall quadratics etc.

I know about vectors yes, I have learned sine cosine laws, somethings about unit circle, area of triangle using 1/2ab Sin C etc.
Even if it's something advanced I'm open to it, you don't have to teach me personally, if there is a link or formula I have no problem experimenting on my own, in fact I tried something random and ended up with 59 degrees, which is close to the answer they got, I pretended that the longitudes were the same, and used the latitude given 63 degrees, as the sector angle, then used the formula for finding the length of a arc of longitude... so just playing around with it until I understand, that's how I learn.

7. I just took a closer look at the problem, and realized that it isn't as different as it looks. Here it is again:

Now for the second part of the question, this I'm stuck like Ive never been before.
''The shortest distance between Town P (63° N,70° E) and Town Q (y° N, 110° W) is 6702.9km
Determine y° N, the latitude of Q.
Yes, the longitudes are different, but what are they? By how much do they differ? What does that tell you about the great circle joining them?

I fell victim to a common error among students: seeing the superficial features of a problem and assuming it requires a complicated method, rather than just trying to do something, and so discovering what it really is.

8. Originally Posted by Dr.Peterson
I just took a closer look at the problem, and realized that it isn't as different as it looks. Here it is again:

Yes, the longitudes are different, but what are they? By how much do they differ? What does that tell you about the great circle joining them?

I fell victim to a common error among students: seeing the superficial features of a problem and assuming it requires a complicated method, rather than just trying to do something, and so discovering what it really is.
They differ by 40 degrees, I would guess it tells me that they're on the greenwich meridian circle?

9. Originally Posted by richiesmasher
They differ by 40 degrees, I would guess it tells me that they're on the greenwich meridian circle?
No, they do not differ by 40 degrees! Look again: they are 70° E and 110° W. What is the difference? (Think of it as positive and negative.) I failed to see this at first, too ...

And neither is the Greenwich meridian, which is 0°.

10. Originally Posted by Dr.Peterson
No, they do not differ by 40 degrees! Look again: they are 70° E and 110° W. What is the difference? (Think of it as positive and negative.) I failed to see this at first, too ...

And neither is the Greenwich meridian, which is 0°.
Well right, one is positive and one is negative... so it tells me they're on the same circle perhaps?

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