Although Hipparchus of Nicaea (c. 190 – c. 120 BC) invented trigonometry, he did not use the three trigonometric ratios sine, cosine, and tangent commonly used today [1]. Hipparchus used the chord ratio instead (Figure 1). Determine the equivalent expression for the chord ratio in terms of the sine ratio.

[1] Aryabhata (476 – 550 AD) invented the sine and cosine, and Yi Xing (683 – 727 AD) calculated the first table of tangents.

【Solution】

According to Figure 2,

Since ,

Write

Since