# Determining Lat, Long using trigonometry

• Insanity
In summary, the conversation discusses the possibility of determining the latitude and longitude coordinates for a specific location on Earth using the known positions of the True North Pole and the Geomagnetic North Pole. The method involves solving a spherical triangle using standard identities of spherical trigonometry, with one side having a known length and the other two sides and angles being determined through calculations. It is suggested that this method could potentially lead to determining the coordinates of the desired location.
Insanity
I am not sure if this is entirely possible, as my spherical trig skills are lacking.

Would it be possible to determine the Lat and Long coordinates for a position on the Earth knowing the following.

1. True North Pole is at 90°N
2. Geomagnetic North Pole is at 82.7°N, 114.4°W
3. Knowing the direction of True North, I can determine the angle separating the two at my position.

The vertices of the spherical triangle would be.
1. Point A would be the True North Pole.
2. Point B would be the Geomagnetic North Pole.
3. Point C would be my location.

The sides of the spherical triangle would be.
1. Line AB, or γ
2. Line BC, or α
3. Line AC, or β

Side γ has a known length, the distance from the True Pole to GM Pole, which is 820.76km. With Earth's radius at 6371km, this works out to be 0.13 radians, or 7.38°.

$sin C/sin γ = -1.07703 = sin A/sin α = sin B/sin β$ Correct?

Seems to me the other lengths should be able to be determined, and then perhaps the coordinates of Point C. Is that possible?

Hi Insanity!

(i'm not sure what your knowns are, but anyway …)

You can solve any spherical triangle by using the standard identities of spherical trigonometry …

Insanity said:
Side γ has a known length, the distance from the True Pole to GM Pole, which is 820.76km. With Earth's radius at 6371km, this works out to be 0.13 radians, or 7.38°.

$sin C/sin γ = -1.07703 = sin A/sin α = sin B/sin β$ Correct?

Maybe I have the law of sines upside down?

C is the angle between vertices, γ is the length of the opposite side in degrees, so then

sin γ/sin C = -0.92848

I know the lat, long of two vertices, A and B, of the triangle, and the length of one side γ, and the angle of its opposite vertice C.

## 1. How is trigonometry used to determine latitude and longitude?

Trigonometry is used to determine latitude and longitude by using the principles of geometry and the relationships between angles and sides of a triangle. By measuring the angle between the horizon and a celestial object, such as the sun or a star, and using the known distance between the observer and the object, the latitude and longitude can be calculated.

## 2. Can trigonometry be used to determine latitude and longitude on Earth?

Yes, trigonometry can be used to determine latitude and longitude on Earth. The principles of trigonometry are applicable to any triangle, including the imaginary triangles formed by the Earth's surface and the lines of longitude and latitude.

## 3. What tools are needed to determine latitude and longitude using trigonometry?

To determine latitude and longitude using trigonometry, an observer will need a device to measure the angle between the horizon and a celestial object, such as an astrolabe or sextant, as well as knowledge of the distance between the observer and the object. Additionally, a map or chart of the area may be helpful in plotting the coordinates.

## 4. Can trigonometry be used to determine latitude and longitude without using celestial objects?

Yes, trigonometry can also be used to determine latitude and longitude without using celestial objects. By measuring the angle between the horizon and two known landmarks, such as two mountain peaks, and using the known distance between the landmarks, the latitude and longitude can be calculated using trigonometry.

## 5. How accurate is determining latitude and longitude using trigonometry?

The accuracy of determining latitude and longitude using trigonometry will depend on the precision of the measurements and calculations. With careful observations and accurate data, trigonometry can provide a high level of accuracy in determining latitude and longitude, often within a few miles or less.

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