已知α，β都是锐角，且tanα＝43，cos(α+β)＝−1114，则β的值是π3π3．

Question

已知α，β都是锐角，且tanα＝4√(3)，cos(α+β)＝−(11)/(14)，则β的值是

(π)/(3)

(π)/(3)

．

Answer 1

∵α，β都是锐角，cos(α+β)＝−(11)/(14)，∴sin（α+β）=√(1−cos2(α+β))=(5√(3))/(14)．
∴tan（α+β）=(sin(α+β))/(cos(α+β))=−(5√(3))/(11)．
∴tanβ=tan[（α+β）-α]=(tan(α+β)−tanα)/(1+tan(α+β)tanα)=(−(5√(3))/(11)−4√(3))/(1−(5√(3))/(11)×4√(3))=√(3)
∵β是锐角，∴β=(π)/(3)．
故答案为(π)/(3)．