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sin3x sinx

解法一:等价无穷小 lim sinx/sin3x x→0 =lim x/(3x) x→0 =⅓ 解法二:洛必达法则 lim sinx/sin3x x→0 =lim cosx/(3cos3x) x→0 =cos0/(3·cos0) =1/(3·1) =⅓

sin3x = sin(x+2x) =sinx*cos2x + cosx*sin2x =sinx*( cosx^2 - sinx^2) + 2 * cosx*sinx*cosx =sinx*cosx^2 - sinx^3 + 2*sinx*cosx^2 =3*sinx*cosx^2 - sinx^3 (像这种题目,应该放在一道题目中,不然展开是没什么用的;)

sin3x=3sinx-4sin³x, cos3x=4cos³x-3cosx,

sin3x =sin(2x+x) =sin2x×cosx+cos2x×sinx =2×sinx×cosx×cosx+(1-2sin²x)×sinx =2sinx×(1-sin²x)+(1-2sin²x)×sinx =2sinx×(1-sin²x)+(1-2sin²x)×sinx =2sinx-2sin³x+sinx-2sin³x=3sinx-4sin³x

首先,利用两次积化和差公式: sinXsin2Xsin3X =-(1/2)(cos3X-cosX)sin3X =-1/4(sin6X)+1/2(sin4X)+1/2sin(2X) 分别设u1,u2,u3为-1/4(sin6X),1/2(sin4X),1/2sin(2X) 则u1的n阶导数为-1/4(sin(6X+n(π/2))*6^(n).....这个是复合函数求导 同理u2的n阶...

lim(x->0) (3sinx - sin3x) /x^3 (0/0) =lim(x->0) (3cosx - 3cos3x) /(3x^2) (0/0) =lim(x->0) (-3sinx + 9sin3x) /(6x) (0/0) =lim(x->0) (-3cosx + 27cos3x) /6 =24/6 =4

利用积化和差公式-2sin((A+B)/2)*sin((A-B)/2)=cosA-cosB 2sin(x/2)*sinx=cos(x/2)-cos(3x/2) 2sin(x/2)*sin2x=cos(3x/2)-cos(5x/2) ... 2sin(x/2)*sinnx=cos((2n-1)x/2)-cos((2n+1)x/2) 裂项相消 原式就等于cos(x/2)-cos((2n+1)x/2)

n=1时公式成立; 现在假设对n-1公式成立 那么sinx+sin2x+sin3x+……+sinnx=sinx+sin2x+sin3x+……+sin(n-1)x+sinnx =[sin((n-1)x/2)sin(nx/2)]/sin(x/2)+sinnx =[sin((n-1)x/2)sin(nx/2)+sinnxsin(x/2)]/sin(x/2) =sin(nx/2)[sin((nx/2-x/2)+2cos(nx...

这题利用公式求

证明 分子 sin(4x+3x)+sin(4x-3x) =sin4xcos3x+cos4xsin4x+sin4xcos3x-cos4xsin4x =2sin4xcos3x 分母 sin(4x+x)+sin(4x-x) =sin4xcosx+cos4xsinx+sin4xcosx-cos4xsinx =2sin4xcosx 左=(2sin4xcos3x)/(2sin4xcosx)=cos3x/cosx,其中 ...

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