1.利用万能公式,我试算了一下,前一半好算,后一半非常之麻烦,不知数字有没错?
sina=2tan(a/2)/[1+tan^2(a/2)],令,tan(a/2)=t,(方便后面运算),则有
4/5=2t/(1+t^2),
2t^2-5t+2=0,t1=1/2,t2=2(不合,舍去,α,β都是锐角).
t=1/2,
cosa=(1-t^2)/(1+t^2)=3/5.
cos(α+β)=5/13,
cosa*cosβ+sina*cosβ=5/13,
cosa+sina*tanβ=5/(13cosβ),
令,tan(β/2)=n,
3/5+4/5*(2n)/(1-n^2)=5(1+n^2)/[13(1-n^2)]
化简得32n^2-52n-7=0,
(4n-7)(8n+1)=0,
n1=7/4(不合,舍去),n2=-1/8(不合,舍去).
tan(β/2)=7/4>1,大于1就等于大于45度啊,β>90.
方程无解啊?
2.α,β都是锐角,tan α =1/7,sinβ=(√10)/10
cosβ=√(1-sin^2β)=3√10/10.
tanβ=sinβ/tanβ=1/3,
tan2β=2tanβ/[1-tan^2(β)]=3/4.
tan(α+2β)=[tana+tan2β]/[1-tana*tan2β]
=(1/7+3/4)/(1-1/7*3/4)
=1.
3.化简：
（1）1/sin10°- (√3)/cos10°
=[1/2cos10-(√2/2)*cos10]/(2sin10*cos10)
=(sin30*cos10-cos30*sin10)/sin20
=sin20/sin20
=1.
（2）sin40°(tan10°-√3)
=2sin40(1/2*sin10-√3/2*cos10)/cos10
=2sin40(sin10*cos60-cos10*sin60)/sin(90-10)
=-2sin40*sin50/(2sin40*cos40)
=-sin50/cos40
=-1.
（3）tan70°cos10°[(√3)tan20°-1]
=2tan70*cos10[(√3/2)sin20-1/2*cos20]/cos20
=2tan70*cos10*[sin20*cos30-cos20*sin30]/cos20
=-2sin10*cos10*tan70/cos20
=-tan20*tan(90-20)
=-tan20*ctn20
=-1.
（4）sin50°[1+(√3)tan10°]
=2sin50*[1/2*cos10+√3/2*sin10]/cos10
=2sin50*[sin30*cos10+cos30*sin10]/cos10
=2sin50*sin(90-50)/cos10
=2sin50*cos50/cos10
=sin100/cos10
=sin80/cos10
=sin(90-10)/cos10
=1.