Throughout the Islamic golden age (9th-13th centuries AD), Muslims would make great contributions to the field of trigonometry. In particular, Muslims would use their observations in astronomy to accomplish tasks, such as facing Mecca, that would require an incredible amount of precision. This level of accuracy could not be calculated by the rudimentary trigonometry of the time, however, through the development of spherical trigonometry, Muslims could pinpoint the exact location of Mecca from any point on the globe.
During the Graeco-Arabic Translation movement (7th-8th centuries AD), Muslims actively translated knowledge on trigonometry developed by past civilizations such as the ancient Greeks and Romans. From this, Muslims derived the ability to create rough estimations of the location of Mecca. First, Muslims would determine the change in longitude (Δλ) and latitude (Δφ) between themselves and Mecca. This would be possible through known procedures developed by Eratosthenes and Ptolemy. These changes could be interpreted as distance, forming the adjacent and opposite sides of a right triangle. Next, using a tangent function, Muslims would calculate the needed angle to face Mecca. Finally, in order to know what orientation to face and direction to turn in, Muslims would need to know their triangles’ location relative to Mecca. By simply using the cardinal directions west-east as an x-axis and north-south as an y-axis, Muslims would form a coordinate plane, (with east and north being positive). Depending on their quadrant, they would position themselves accordingly. Quadrants 2 and 4 would require Muslims to turn in a positive/counterclockwise direction while quadrants 1 and 3 would require the opposite. For example, Muslims southeast of Mecca residing in the fourth quadrant would turn in a positive direction. On the contrary, Muslims northeast of Mecca in the first quadrant would turn in a negative direction.
The Image above is an example of this application. To solve for angle T (marked in red) from the city of Tehran, you would first determine Δλ for side t, followed by Δφ for side m. Using this information, you would use a Tangent function to determine angle T. The resulting equation would be tan(T) = Δλ/Δφ or T° = arctan(Δλ/Δφ). Finally, after accounting for your location in quadrant 1, you would orientate yourself south and turn in a negative/clockwise direction for angle T.
While this method using basic trigonometry gave a rough estimation of Mecca’s location, Muslims soon found the method to be inaccurate as it did not account for the curvature of the Earth. Eventually, Muslim mathematician Al Burini developed the equation q = arctan[(sinΔλ)/(cosφd*tanφm-sinφd*cosΔλ)], with new terms such as φd being the latitude of your location and φm the latitude of Mecca. He combined this equation with astronomical observations of lunar eclipses and of the zenith of the sun to fully account for the Earth’s curvature. This development finally gave Muslims across the globe the ability to face Mecca with remarkable precision. These calculations developed by medieval Muslims would go on to heavily advance the field of spherical trigonometry, which is still used in many modern technologies such as GPS.
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