i) If 2sin2x + cos2x = k, where k is a constant and k not = -1, show that
(1+k)tan^2 x - 4tanx + k-1 =0
ii) Hence, show that if tanx1 and tanx2 are the roots of this quadratic equation in tanx, then tan(x1 +x2) =2
Trigonometry - compound angles
2008-03-09 7:14 pm
回答 (1)
2008-03-09 7:30 pm
✔ 最佳答案
i) 2 sin 2x + cos 2x = k 4 sin x cos x + 2 cos2 x - 1 - k = 0
Dividing by cos2 x,
4 tan x + 2 - ( k + 1 )( 1 + tan2 x ) = 0
( k + 1 ) tan2 x - 4 tan x + k + 1 - 2 = 0
( k + 1 ) tan2 x - 4 tan x + ( k - 1 ) = 0
ii) tan x1 + tan x2 = 4 / ( k + 1 )
tan x1 tan x2 = ( k - 1 ) / ( k + 1 )
tan ( x1 + x2 )
= ( tan x1 + tan x2 ) / ( 1 - tan x1 tan x2 )
= [ 4 / ( k + 1 ) ] / [ 1 - ( k - 1 ) / ( k + 1 )]
= 4 / ( k + 1 - k + 1 )
= 4 / 2
= 2
參考: My Maths Knowledge
收錄日期: 2021-04-14 19:51:30
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