欢迎光临散文网 会员登陆 & 注册

[Geometry] Chord Ratio

2021-07-05 12:18 作者:AoiSTZ23  | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

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.

Figure 1

【Solution】

Figure 2

According to Figure 2,

chord%20(%5Ctheta)%20%3D%5Csqrt%7B(1-%5Ccos%20%5Ctheta%20)%5E2%2B%5Csin%5E2%E2%81%A1%20%5Ctheta%7D%0A

 chord%20(%5Ctheta)%20%3D%5Csqrt%7B1-2%5Ccos%20%5Ctheta%20%2B%5Ccos%5E2%20%5Ctheta%20%2B%20%5Csin%5E2%E2%81%A1%20%5Ctheta%7D

Since %5Ccos%5E2%E2%81%A1%20%5Ctheta%20%2B%20%5Csin%5E2%E2%81%A1%20%5Ctheta%3D1,

chord(%CE%B8)%3D%5Csqrt%7B2-2%20%5Ccos%E2%81%A1%20%5Ctheta%7D

Write

chord(%CE%B8)%3D2%5Csqrt%7B%5Cfrac%7B1-%5Ccos%E2%81%A1%20%5Ctheta%7D%7B2%7D%7D

Since %5Csqrt%7B%5Cfrac%7B1-%5Ccos%E2%81%A1%20%5Ctheta%7D%7B2%7D%7D%20%3D%20%5Csin%20%5Cfrac%7B%5Ctheta%7D%7B2%7D%EF%BC%8C

chord(%5Ctheta)%3D2%20%5Csin%E2%81%A1%5Cfrac%7B%CE%B8%7D%7B2%7D

 


[Geometry] Chord Ratio的评论 (共 条)

分享到微博请遵守国家法律