[Trigonometry] Triple-angle Sine

2021-10-08 09:54 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

In the "Treatise on the Chord and Sine", the Persian astronomer and mathematician Jamshid al-Kashi (1380 - 1429) used the trigonometric identity $%5Csin(3x)%20%3D%203%5Csin(x)%20-%204%5Csin%5E3(x)$ to calculate $%5Csin%20%7B1%7D%5E%7B%5Ccirc%7D%20$ accurate to 18 decimal places ( $%5Csin%20%7B1%7D%5E%7B%5Ccirc%7D%20%3D%200.017452406437283571$ ). Use the trigonometric identities $%5Csin(2x)%20%3D%202%5Csin(x)%5Ccos(x)$ and $%5Ccos(2x)%20%3D%201-2%5Csin%5E2(x)%20$ to prove that $%5Csin(3x)%20%3D%203%5Csin(x)%20-%204%5Csin%5E3(x)$ .

【Solution】

Use the angle sum formula $%5Csin(x%2By)%20%3D%20%5Csin(x)%5Ccos(y)%20%2B%20%5Ccos(x)%5Csin(y)$ and make the substitution $y%20%3D%202x$ .

$%5Csin(x%2B2x)%20%3D%20%5Csin(x)%5Ccos(2x)%20%2B%20%5Ccos(x)%5Csin(2x)$

Substitute the double-angle formulas for sine and cosine into the above formula:

$%5Csin(2x)%20%3D%202%5Csin(x)%5Ccos(x)$

$%5Ccos(2x)%20%3D%201-2%5Csin%5E2(x)$

Subsequently,

$%20%5Csin(x%2B2x)%20%3D%20%5Csin(x)%20%5Ccdot%20%5Cleft%5B1-2%5Csin%5E2(x)%5Cright%5D%20%2B%20%5Ccos(x)%20%5Ccdot%20%5Cleft%5B2%5Csin(x)%5Ccos(x)%5Cright%5D$