13、化简：（1）3√15sinx+3√5cosx （2）3/2cosx-√3/2sinx （3）√3sinx/2+cosx/2

题目

13、化简：（1）3√15sinx+3√5cosx （2）3/2cosx-√3/2sinx （3）√3sinx/2+cosx/2
（4）√2/4sin(π/4-x)+√6/4cos(π/4-x) （5）sin347°cos148°+sin77°cos58°
（6）sin164°sin224°+sin254°sin314° （7）sin(a+b)cos(γ-b)-cos(b+a)sin(b-γ)
（8）sin(a-b)cos(b-γ)-cos(a-b)sin(γ-b)
（9）tan5π/4+tan5π/12 / 1-tan5π/12
（10）sin(a+b)-2sinacosb) / 2sinasinb+cos(a+b)

答案

1）3√15sinx+3√5cosx
=6√5(√3/2sinx+1/2cosx )
=6√5sin(x+π/3)
2）3/2cosx-√3/2sinx
=√3(√3/2cosx-1/2sinx )
=√3cos(x+π/6)
3）√3sinx/2+cosx/2
=2(√3/2sinx/2+1/2cosx/2 )
=2sin(x/2+π/6)
4）√2/4sin(π/4-x)+√6/4cos(π/4-x)
=√2/2[1/2sin(π/4-x)+√3/2cos(π/4-x) ]
=√2/2sin(5π/12-x)
5）sin347°cos148°+sin77°cos58°
=sin13°cos32°+cos13°sin32°
=sin50°
6）sin164°sin234°+sin254°sin314°(题有误应该是sin234°)
=-sin16°sin54°+sin74°sin46°
=-sin16°sin54°+cos16°cos54°
=cos70°
7）sin(a+b)cos(γ-b)-cos(b+a)sin(b-γ)
＝sin(a+b)cos(γ-b)+cos(b+a)sin(γ-b)
=sin(a+b+γ-b)
=sin(a+γ)
8）sin(a-b)cos(b-γ)-cos(a-b)sin(γ-b)
=sin(a-b)cos(γ-b)-cos(a-b)sin(γ-b)
=sin(a-b-γ+b)
=sin(a-γ)
9）tan5π/4+tan5π/12 / 1-tan5π/12
=tanπ/4+tan5π/12 / 1-tan5π/12
=(tanπ/4+tan5π/12) / (1-tanπ/4*tan5π/12 )
=tan(π/4+5π/12)
=tan2π/3
=-√3
10）sin(a+b)-2sinacosb) / 2sinasinb+cos(a+b)
=[sinacosb+cosasinb-2sinacosb]/[cosacosb-sinasinb+ 2sinasinb]
=(cosasinb-sinacosb)/(cosacosb+sinasinb)
=sin(b-a)/cos(b-a)
=tan(b-a)