Question

已知向量a=(cosa,sina),b=(cosβ,-sinβ)，|a-b|=2倍根号5/5

已知向量a=(cosa,sina),b=(cosβ,-sinβ)，|a-b|=2倍根号5/5

Answer 1

(I)
a-b = (cosα-cosβ, sinα+sinβ)

|a-b|^2 =(cosα-cosβ)^2+(sinα+sinβ)^2
(cosα-cosβ)^2+(sinα+sinβ)^2 = 4/5
2 -2cos(α+β)=4/5
cos(α+β) = 3/5
(II)
sinβ =5/13 =>cosβ =12/13

cos(α+β) = 3/5
cosα.cosβ -sinα.sinβ = 3/5
(12/13)cosα - (5/13)sinα = 3/5
60cosα - 25sinα = 39
60cosα = 25sinα+ 39
3600(cosα)^2 =(25sinα+ 39)^2
4225(sinα)^2 +1950sinα -2079=0
sinα = (-1950+6240)/8450
=4290/8450
=429/845