Question

高中三角函数题两道：1.SinA+SinB=a,CosA+CosB=1+a,求Sin（A+B），Cos（A+B）。

2.基本相同：SinA+CosB=二分之根三,CosA+SinB=根二,求TanA*CotB
我意思是题型差不多。

Answer 1

1.SinA+SinB=a,CosA+CosB=1+a,求Sin（A+B），Cos（A+B）。
解：sinA+sinB=a, 平方之得 sin²A+2sinAsinB+sin²B=a².................... (1)
cosA+cosB=1+a, 平方之得 cos²A+2cosAcosB+cos²B=(1+a)²............(2)
(1)+(2)得cos(A-B)=[a²+(1+a)²-2]/2=a²+a-1/2..................(3)
(2)-(1)得2cos(A+B)+cos2A+cos2B=2cos(A+B)+2cos(A+B)cos(A-B)
=2cos(A+B)[1+cos(A-B)]=2a+1
故cos(A+B)=(2a+1)/2[1+cos(A-B)]=(2a+1)/2(1+a²+a-1/2)=(2a+1)/(2a²+2a+1)
(sinA+sinB)(cosA+cosB)=sinAcosA+sinBcosB+sinAcosB+cosAsinB
=(1/2)(sin2A+sin2B)+sin(A+B)=sin(A+B)cos(A-B)+sin(A+B)
=sin(A+B)[cos(A-B)+1]=sin(A+B)(a²+a+1/2)=a(1+a)
∴sin(A+B)=a(1+a)/(a²+a+1/2)=2a(1+a)/(2a²+2a+1).
2.sinA+cosB=√3/2, ,cosA+sinB=√2, 求tanA*cotB
解：sinA+cosB=√3/2,平方之得 sin²A+2sinAcosB+cos²B=3/4..............(1)
cosA+sinB=√2,平方之得 cos²A+2cosAsinB+sin²B=2...........................(2)
(1)+(2)得 sin(A+B)=3/8.......................................................................(3)
(1)-(2)得 2sin(A-B)-(cos2A-cos2B)=2sin(A-B)+2sin(A+B)sin(A-B)
=2sin(A-B)[1+sin(A+B)]=2sin(A-B)(1+3/8)=(11/4)sin(A-B)=-5/4
故sin(A-B)=-5/11...............................................................................(4)
tanAcotB=tanA/tanB=[sin(A+B)-sin(A-B)]/[sin(A+B)+sin(A-B)]
=(3/8+5/11) /(3/8-5/11)=-73/7

Answer 2

1、sinA+sinB=a(1)
cosA+cosB=1+a(2)
(1)²+(2)²
sin²A+sin²B+2sinAsinB+cos²A+cos²B+2cosAcosB=a²+a²+2a+1
2+2cos(A-B)=2a²+2a+1
cos(A-B)=a²+a-1/2
(2)²-(1)²
cos2A+cos2B+2cosAcosB-2sinAsinB=2a+1
cos(A+B+A-B)+cos[A+B-(A-B)]+2cos(A+B)=2a+1
cos(A+B)cos(A-B)-sin(A+B)sin(A-B)+cos(A+B)cos(A-B)+sin(A+B)sin(A-B)+2cos(A+B)=2a+1
2cos(A+B)cos(A-B)+2cos(A+B)=2a+1
cos(A+B)[cos(A-B)+1]=a+1/2
cos(A+B)(a²+a-1/2+1)=a+1/2
cos(A+B)=(a+1/2)/(a²+a+1/2)
cos(A+B)=(2a+1)/(2a²+2a+1)
由(1)
sin(A+B-B)+sin(A+B-A)=a
sin(A+B)cosB-cos(A+B)sinB+sin(A+B)cosA-cos(A+B)sinA=a
sin(A+B)(cosA+cosB)-cos(A+B)(sinA+sinB)=a
sin(A+B)(1+a)=a+(2a+1)a/(2a²+2a+1)
sin(A+B)(1+a)=a(2a²+2a+1+2a+1)/(2a²+2a+1)
sin(A+B)(1+a)=2a(a+1)²/(2a²+2a+1)
sin(A+B)=2a(a+1)/(2a²+2a+1)

Answer 3

1.sin(a+b)=2tan[(a+b)/2]/{1+tan^2[(a+b)/2]}=2a(1+a)/(1+2a+2a^2)
cos(a+b)={1-tan^2[(a+b)/2]}/{1+tan^2[(a+b)/2]}=(1+2a)/(1+2a+2a^2)
2.基本相同是什么地方相同？已知条件吗？