Explanation:
See image
tan x = 4/5 and cos x less than zero means sin x is less than zero too because tan x = sinx / cosx <=== ( in Q III)
sin / cos = 4/5
cross multiply to get
5 sin = 4 cos square both sides
25 sin^2 = 16 cos^2 remember: sin^2 + cos^2 =1 then cos^2 = 1 -sin^2 :
25 sin^2 = 16 (1-sin^2)
41 sin^2 = 16
sin^2 = 16 / 41 now square root both sides
sin = ± 4 / sqrt(41)
BUT remember...sin x is negative for this QIII angle: sin x = - 4/sqrt(41)
or sin x = - 4 sqrt (41) / 41