Spherical Trigonometry

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Spherical Trigonometry

• Finding the qibla (direction for prayer) for any
  location in the world.

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Step 1 Find the latitude and longitude for your
city.
Well use Minneapolis as our example. Its
coordinates are 4458 N Latitude and Mecca
21º27 N Latitude and 9316 W. Longitude
39º49 E Longitude You can use the websites
below to find other coordinates http//www.astro.
com/cgi-bin/atlw3/aq.cgi?lange (you can enter
city names) http//www.realestate3d.com/gps/world-
latlong.htm (list of various cities)
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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!!
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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.
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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 - 4458 452 a 90
- 2127 6833 N 9316 3949 1335
North Pole (90 latitude)
Equator (0 latitude)
Prime Meridian (O longitude)
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Here is a view of our situation from the directly
above the North Pole.
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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 (452) cos
(6833) sin (452) sin (6833) cos
(1335) cos n -0.1913 n 101.03
In order to find the actual distance, we convert
this to radians 101.03 p/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.
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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!!