Question

sin(α+β)=5/13 sinα=4/5 α是锐角 β是钝角 求cosβ

Answer 1

sin(α+β)=5/13 sinα=4/5 α是锐角 β是钝角 求cosβ

首先：cosα= 3/5

sin(α+β)= sinαcosβ + cosαsinβ = 4/5cosβ + 3/5sinβ = 5/13

设t = cosβ 有 4t/5 + 3根号(1-t^)/5 = 5/13

有 3根号(1-t^)/5 = 5/13 - 4t/5

两边平方得：9(1-t^)/25 = ( 5/13 - 4t/5 )^

整理得：t^ - 8t/13 + 25/169 - 9/25 = 0

直接用求根公式得 t = - 16/65 或 56/65(不符合钝角）

所以 cosβ = - 16/65