Like countless other cultures, the development of trigonometry in Indian culture stemmed from its significantly important purposes in astronomy. It is presumed that the basic knowledge of trigonometry developed from Greek knowledge, shared amongst the Roman trade routes. This assumption is further justified due to Ancient Indian mathematicians referring to the standard radius of a circle as "R", just as Hipparchus did. Nonetheless, Indian Mathematicians were rapidly devolving new improvements to previously known identities and laying foundational knowledge for millennia to come.
Figure 1: Illustration of trig ratios in relation to arcs and chords.
From the 4th to 5th centuries, in the scripture, The Siddhantas, sine was defined as the relationship between half an angle and half a chord. Mathematician Arybhata in 499 AD developed a table of values for sine and cosine, using “jya” or “zya” for sine and “kojya” or “kozya” for cosine. "Jya" is a Sanskrit word used to define a "bow-string", which describes what sine visually looks like (see Figure 1). These names were later mistranslated and misheard across the globe, and are employed nowadays as sine and cosine.
To trace back the history of trigonometric identities applied in Mathematics Extension 1, developments leading up to the discovery must be addressed. For instance, the construction of sine tables played an incredibly large role in the development of trigonometric identities, The earliest version of this table was constructed in the 5th century and was known as the “Paitamahasiddhanta”. Nonetheless, the most well-preserved sine table is from Arybhata’s work, Aryabhatiya (see Figure 2). This sine table sparks continuous discourse amongst mathematicians who question the mathematical steps Arybhata took to achieve his result. A common perspective is that the tabular values were achieved by employing the cumulative sine differences to find an approximate sine value. Others argue that the values were achieved from the prior knowledge of the known values of Sin(30), Sin(45), Sin(60), and Sin (90), as well as the trigonometric identities of sin2x+cos2x=1 (since R=1).
Figure 2: Table of Sine values
From this table, the sum of angles as an identity was formed. In 628 AD mathematician, Brahmagupta found a method through which he could find the value of sin (A + x), in which A is a known tabulated value and x lies between 0-225. At this point, the identity relied on the differences in the tabulated values and was extremely complex. Nonetheless, he kept adjusting his identity so that he eventually achieved the identity below (Figure 3).
Figure 3: Brahmagupta's identity for sine (a+x)
From this, in 1150 AD, Bhaskaracharya, otherwise known as Bhaskara II, produced his famous text Siddhanta Shrimoni in which, he was able to formulate the identities we utilise today of sin(a + b) = sin a cos b + cos a sin b and sin(a – b) = sin a cos b – cos a sin b. The process he undertook to reach this was incredibly complex, but worthwhile as we still utilise the identity nowadays in Mathematics Extension 1 its applications.
Indian astronomers applied trigonometry significantly in their studies of the stars, planets and moon. For instance, as seen in Figure 5, they applied trigonometric ratios to find the relative distances between the Moon, Sun and Earth. This was done when the moon was half-full so that a right angle triangle could be applied. By doing so they discovered that the angle between the Moon, Sun and Earth was 1/7 degrees.
Figure 4: Example of application of trigonometry in Ancient India