Question

高一三角函数简单化简问题

谢谢了,50分送上！利用和差角公式化简：1.3√15sinx+3√5cosx2.√2(sinx-cosx)3.√3sinx+cosx4.(√2/4)sin(π/4-x)+√6/4cos(π/4-x)5.sin347°cos148°+sin77°cos58°6.sin164°sin224°+sin254°sin314°后面这题请帮我验证下cos(a-b)cos(b-r)-sin(a-b)sin(b-r)化简是否等于cos(a-r)? 

Answer 1

1.3√15sinx+3√5cosx2=6√5（√3/2sinx+1/2cosx)=6√5(sinxcosπ/6+cosxsinπ/6)=6√5sin(x+π/6)
2.√2(sinx-cosx)=2(√2/2sinx-√2/2cosx)=2(sinxcosπ/4-cosxsinπ/4)=2sin(x-π/4)
3.√3sinx+cosx=2(√3/2sinx+1/2cosx)=2(sinxcosπ/6+cosxsinπ/6)=2sin(x+π/6)
4.(√2/4)sin(π/4-x)+√6/4cos(π/4-x)=√2/2(1/2sin(π/4-x)+√3/2cos(π/4-x))=√2/2(sin(π/4-x)cosπ/6+cos(π/4-x)sinπ/6=√2/2sin(π/4-x+π/6)=√2/2sin(5π/12-x)=√2/2sin(π/2-(π/12+x))=√2/2cos(π/12+x)
5.sin347°cos148°+sin77°cos58°=sin13°cos32°+sin(45°+32°)cos(13°+45°)
sin(45°+32°)=sin45°cos32°+cos45°sin32°=√2/2(cos32°+sin32°)
cos(13°+45°)=cos13°cos45°-sin13°sin45°=√2/2(cos13°-sin13°)
sin(45°+32°)cos(13°+45°)=1/2（cos32°cos13°-cos32°sin13°+sin32°cos13°-sin13°sin32°）
sin13°cos32°+sin(45°+32°)cos(13°+45°)=1/2（cos32°cos13°+cos32°sin13°+sin32°cos13°-sin13°sin32°）=1/2(（cos32°cos13°-sin13°sin32°）+（cos32°sin13°+sin32°cos13°））=1/2（cos45°+sin45°）=√2/2
6.sin164°sin224°+sin254°sin314°=sin（180°-16°）sin(270°-46°）+sin(270°-16°）sin(360°-46°）=-sin16°cos46°+cos16°sin46°=sin(46°-16°)=sin30°=1/2

另外cos(a-b)cos(b-r)-sin(a-b)sin(b-r)=cos[(a-b)+(b-r)]=cos(a-r)

Answer 2

cos(a-b)cos(b-r)-sin(a-b)sin(b-r)=cos[(a-b)+(b-r)]=cos(a-r)

Answer 3

ggg