Spherical Trigonometry

1. Spherical Trigonometry Finding the qibla (direction for prayer) for any location in the world.

2. Step 1: Find the latitude and longitude for your city. We’ll use Minneapolis as our example. Its coordinates are: 44˚58’ N Latitude and Mecca 21º27’ N Latitude and 93˚16’ W. Longitude 39º49’ E Longitude You can use the websites below to find other coordinates: http://www.astro.com/cgi-bin/atlw3/aq.cgi?lang=e (you can enter city names) http://www.realestate3d.com/gps/world-latlong.htm (list of various cities)

3. Step 2: Create a spherical triangle with your city, Mecca, and the North pole as vertices. Remember that lines on a sphere are Great Circles!!

4. Spherical Law of Cosines: cos n = cos m cos a + sin m sin a cos N N Spherical Law of Sines: m a A n M The sides are measured as arcs of central angles of the sphere.

5. Step 3: Use the latitude and longitude to find 2 sides and the included angle of your spherical triangle. The sides will be measured as arcs. In our example: m = 90˚ - 44˚58’ = 45˚2’ a = 90˚ - 21˚27’ = 68˚33 N = 93˚16’ + 39˚49’ = 133˚5’ North Pole (90˚ latitude) Equator (0˚ latitude) Prime Meridian (O˚ longitude)

6. Here is a view of our situation from the directly above the North Pole.

7. Step 4: Use the Spherical Law of Cosines to find the distance between your city and Mecca. cos n = cos m cos a + sin m sin a cos N so in our example: cos n = cos (45˚2’) cos (68˚33’) + sin (45˚2’) sin (68˚33’) cos (133˚5’) cos n = -0.1913 n = 101.03˚ In order to find the actual distance, we convert this to radians: 101.03˚ • π/180 = 1.7633. Then, because the radius of Earth is approximately 3960 miles, the distance between Minneapolis and Mecca is (1.7633 * 3960) = 6983 miles.

8. Step 5: Use the Spherical Law of Sines to find the initial heading from your city to Mecca. so in our case: and sin A = 0.6926 A = 43.8˚ so we would face 43.8˚ East of North for our initial heading. This is almost directly North-East -- not what you would expect for a city that is South-East of us!!