Question

sina+sinb=2sin1/2(a+b)cos1/2(a+b)

Answer 1

问：sina+sinb=2sin1/2(a+b)cos1/2(a+b)

答：答案为：sin(a) + sin(b) = 2sin(1/2(a+b))cos(1/2(a+b))这是正弦和公式的一个特例。根据正弦和公式，有：sin(a + b) = sin(a)cos(b) + cos(a)sin(b)将a和b分别替换成1/2 a和1/2 b，得到：sin(1/2 a + 1/2 b) = sin(1/2 a)cos(1/2 b) + cos(1/2 a)sin(1/2 b)由于sin(x) = 2sin(1/2x)cos(1/2x)，所以可以将上式改写为：2sin(1/2 a)cos(1/2 a) = 2sin(1/2 b)cos(1/2 b)移项并约分，可得：sin(a) + sin(b) = 2sin(1/2 a)cos(1/2 a) + 2sin(1/2 b)cos(1/2 b)将a和b分别替换成x和y，得到：sin(x) + sin(y) = 2sin(1/2 x)cos(1/2 x) + 2sin(1/2 y)cos(1/2 y)也就是：sin(x) + sin(y) = 2sin(1/2(x+y))cos(1/2(x+y))因此，原式可以化为：sin(a) + sin(b) = 2sin(1/2(a+b))cos(1/2(a+b))其中，a和b可以是任意实数。