在三角行ABC中,sinA+sinC=2sinB,A-C=π/3,求sinB的值要完整过程.
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/01 22:39:54
![在三角行ABC中,sinA+sinC=2sinB,A-C=π/3,求sinB的值要完整过程.](/uploads/image/z/5198245-61-5.jpg?t=%E5%9C%A8%E4%B8%89%E8%A7%92%E8%A1%8CABC%E4%B8%AD%2CsinA%2BsinC%3D2sinB%2CA-C%3D%CF%80%2F3%2C%E6%B1%82sinB%E7%9A%84%E5%80%BC%E8%A6%81%E5%AE%8C%E6%95%B4%E8%BF%87%E7%A8%8B.)
在三角行ABC中,sinA+sinC=2sinB,A-C=π/3,求sinB的值要完整过程.
在三角行ABC中,sinA+sinC=2sinB,A-C=π/3,求sinB的值
要完整过程.
在三角行ABC中,sinA+sinC=2sinB,A-C=π/3,求sinB的值要完整过程.
sinA+sinC=2sin[(A+C)/2]×cos[(A-C)/2]=2sin[(π-B)/2]×cos(π/6)=2sin(π/2-B/2)×√3/2=√3cos(B/2)=√3×(1+cosB)/2.
已知sinA+sinC=2sinB,
则:√3×(1+cosB)/2=2sinB
化简得:cosB=4√3sinB/3-1,将等式两边平方:
则:1-sin²B=(4√3/3sinB-1)²
解得:sinB=8√3/19.
sinA+sinC=2sinB,
和差化积公式,
2sin[(A+C)/2]cos[(A-C)/2]=4sin(B/2)cos(B/2),
sin[(π-B)/2]*cos[(A-C)/2]=2sin(B/2)cos(B/2),
cos(B/2)*cos[(A-C)/2]=2sin(B/2)cos(B/2),......(1)
cos(π/6)=2sin(B...
全部展开
sinA+sinC=2sinB,
和差化积公式,
2sin[(A+C)/2]cos[(A-C)/2]=4sin(B/2)cos(B/2),
sin[(π-B)/2]*cos[(A-C)/2]=2sin(B/2)cos(B/2),
cos(B/2)*cos[(A-C)/2]=2sin(B/2)cos(B/2),......(1)
cos(π/6)=2sin(B/2),.....(2)
sin(B/2)=√3/4,
cos(B/2)=√(1-3/16)=√13/4,
∴sinB=2sin(B/2)cosB/2)
=2*(√3/4)*√13/4.
=√39/8.
收起