急已知C:x^2+(y-1/2)^2=r^2与y=sin(x)的图像有唯一交点,且交点的横坐标为a,则(sin(a)+sin(3已知C:x^2+(y-1/2)^2=r^2与y=sin(x)的图像有唯一交点,且交点的横坐标为a,则(sin(a)+sin(3a)-2(cos^2(a))/cos(a)的值等于

Question

【急】已知C:x^2+(y-1/2)^2=r^2与y=sin(x)的图像有唯一交点,且交点的横坐标为a,则(sin(a)+sin(3
已知C:x^2+(y-1/2)^2=r^2与y=sin(x)的图像有唯一交点,且交点的横坐标为a,则(sin(a)+sin(3a)-2(cos^2(a))/cos(a) 的值等于

Answer 1

C是以(0,1/2)为圆心的圆,圆心在y = sin(x)上方.
若C上有点落在y = sin(x)下方,可知二者至少有两个交点.
于是当C与y = sin(x)恰有一个交点时,二者一定在交点处相切.
由交点横坐标为a,知其纵坐标为sin(a).
与圆心连线的斜率为(sin(a)-1/2)/a,因此圆在该点切线的斜率为a/(1/2-sin(a)).
另一方面,y = sIn(x)在该点切线的斜率为cos(a).
于是a/(1/2-sin(a)) = cos(a),即2a = cos(a)-2sin(a)cos(a) = cos(a)-sin(2a).
sin(a)+sin(3a) = 2sin(2a)cos(a).
因此(sin(a)+sin(3a)-2(cos²(a))/cos(a) = 2(sin(2a)-cos(a)) = -4a.
估计题意就是化到这一步,不会真的去求a的值.
2a = cos(a)-sin(2a)这个方程的解大概是没有办法写出来的.
倒是可以给一个数值解:a = 0.24737421945816...