Trigonometry flourished in Islamic civilizations and became a stand alone discipline, as humans had used trigonometry mostly in relation to astronomy. Indeed, Nasir al-Din aTusi, a mathematician of the 13th century, is credited with establishing trigonometry as an independent field. Below, we take the reader to a (or the) principle discovery and element in trigonometry: the sine law.

3.1 The Plane and Spherical Sine Laws

The plane sine law states that, for any 2-D triangle, we have:

b sin (3) a sin(a) C sin(y)’

where a, b, and c are the sides of the triangle, and a, 3, and y are the opposite angles of the sides, respectively. In laymen terms, the sine law stipulates that the ratio of a side to the opposite angle is equivalent to a ratio of another side and it’s opposite angle. See the below diagram to help illustrate the equation better.

α b a

Figure 1: Plane Triangle. Source:

Now, the spherical sine law is as follows:

sin(a) sin (A) sin (3) sin (B) sin(7) sin(C)

Note that the above holds for a unit sphere, or a sphere with radius 1, and that the triangle is actually curved onto the sphere, instead of on a plane. See the diagram on the next page for an illustration.

3.2 Who Actually Discovered the Sine Laws?

Attributing math discoveries to mathematicians is often a very difficult if not impossible task, since math builds heavily on previous math discoveries and historical evidences can be hard to decipher and date. This is even more so the case in terms of the sine law.

The sine law was “discovered almost simultaneously in the late tenth century by Prince Abu Nasr bin Iraq and Abu Al Wafa’ Al-Buzjani, and considerable controversy ensued over who deserved the credit for being first” [21]. Al-Tusi, who constructed an elegant proof for the sine law, actually credits Al-Khojandi for its discovery, yet this is difficult to believe for reasons discussed in Al-Khojandi’s biography in the next section.

Regardless of who deserves credit, we introduce four relevant Muslim mathematicians below: the three co-discoverers, and Al-Tusi, whose proof we present in Section 3.4.