高一简单的三角函数化简 （正弦sin与余弦cos之间的化简）

Question

机器语言表示：y=5*[sin（x)]^2+[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
("^"代表次方，大家应该都知道的)
自然语言表示：y等于 5倍的sin平方x , 加上根号3倍的sinx乘以cosx，加上6倍的cos平方x
答案给的是：y=5.5+sin(2x+π/6)
请问这个答案是怎么出来的？需要十分详细的步骤！谢谢！！！

Answer 1

y=5*[sin（x)]^2+[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
=5 + 2分之根号3*sin2x + (cosx)^2
=5 + 2分之根号3*sin2x + (cos2x+1)/2
=2分之根号3*sin2x + 二分之一cos2x + 5.5
=sin(2x+π/6) + 5.5

Answer 2

y=y=5*[sin（x)]^2+[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
=5*[sin（x)]^2+6*[cos(x)]^2 +[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
=5*[sin（x)]^2+5*[cos(x)]^2 +[cos(x)]^2 +[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
=5*“【sin（x)]^2+[cos(x)]^2 】 +[cos(x)]^2 +[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
=5 +[cos(x)]^2 +[3^(1/2)]*[sin(x)]*[cos(x)]+6*[cos(x)]^2
=5 +1/2[1+cos(2x)] +√3/2*[sin(2x)]
=5.5 +1/2cos(2x) +√3/2*[sin(2x)]
=5.5 +sin(π/6)cos(2x) + cos(π/6)*[sin(2x)]
=5.5+sin(2x+π/6)

Answer 3

原始=5*[sin（x)]^2+3^(1/2)]*/2sin(2x)+6*[cos(x)]^2
=5+[cos(x)]^2+3^(1/2)]*/2sin(2x)
=5.5+3^(1/2)]*/2sin(2x)+1/2cos(2x)
=5.5+sin(2x+π/6)

Answer 4

解：y=5(sinx)^2+6(cosx)^2+根号3*sinx*cosx=5+(cosx)^2+根号3/2*2sinx*cosx
=5+1/2*(1-sin(2x))+根号3/2*sin(2x)=5.5+根号3/2*sin(2x)-1/2*sin(2x)=5.5+sin(2x+a)
又有a=arctan((1/2)/(根号3/2))=arctan(根号3/3)
所以a=pi/6