Question

tan theta= -4/3 and sec theta<0 find sin (theta = pi/6)

Answer 1

If secØ < 0 then cosØ<o
and the cosine is negative in II and III
given tanØ = -4/3, and the tangent is negative in II and IV
so Ø must be in II

I will assume you want to find sin(Ø + π/6) , (the + sign is on the same key as = , you probably did not hit the shift key)

if tanØ = -4/3 in II, then
sinØ = 4/5, and cosØ = -3/5 , (from the 3-4-5 triangle)

sin(Ø+π/6)
= sinØcosπ/6 + cosØsinπ/6
= (4/5)(√3/2) + (-3/5)(1/2)
= (4√3 - 3)/10