Question

一道数学题，急需，在线等

已知cos(a+π/4)=3/5,π/2<=a<3π/2,求cos(2a+π/4)

Answer 1

cos(a+π/4) = cosa*cosπ/4 - sina*sinπ/4 = (根号2)/2 * (cosa - sina) = 3/5
所以cosa-sina = 3/5 * (根号2)
所以(cosa-sina)^2 = 18/25
所以18/25 = (cosa-sina)^2 = 1 - 2sinacosa = 1-sin2a
解得sin2a = 7/25

因为π/2 <= a <= 3π/2，所以 3π/4 <= a+π/4 <= 7π/4
在这个区间内cos(a+π/4) > 0，所以3π/2 <= a+π/4 <= 7π/4
即5π/4 <= a <= 3π/2
所以5π/2 <= 2a <= 3π
在这个范围内cos2a < 0，根据sin2a = 7/25，得cos2a = -24/25
所以
cos(2a+π/4) = cos2a*cosπ/4 - sin2a*sinπ/4
= -24/25 * (根号2)/2 - 7/25 * (根号2)/2
= -31/50 * (根号2)

Answer 2

∵π/2≤a<3π/2
cos(a+π/4)=3/5
∴a+π/4是第四象限角
∴5π/4＜a＜3π/2

cosacosπ/4-sinasinπ/4=3/5
cosa-sina=3根号2/5
(cosa-sina)^2=(3根号2/5)^2
1-2sinacosa=18/25
2sinacosa=7/25
(cosa+sina)^2=1+2sinacosa=1+7/25=32/25
cosa+sina=-根号(32/25)=-4根号2/5

cos(2a+π/4)=cos2acosπ/4-sin2asinπ/4=根号2/2 (cos2a-sin2a)
=根号2/2 { (cos^2a-sin^2a) -2sinacosa
=根号2/2 { (cosa+sina)(cosa-sina) -2sinacosa }
=根号2/2 { (-4根号2/5)(3根号2/5) -7/25 }
=根号2/2 { -24/25 -7/25 }
=-31根号2/50

Answer 3

ydc

Answer 4

不懂