Under the substitution t = tan(θ/2), the equation 3sinθ + 4cosθ = 5 becomes:

Substitute sinθ = $2t/(1+t^2)$ and cosθ = $(1-t^2)/(1+t^2)$, then multiply through by $1+t^2:6t+4-4t^2=5+5t^2$, which tidies to $9t^2-6t+1=0$. This is $(3t-1{)}^2$, so $t=1/3$ and θ ≈ 36.87°, where $3(0.6)+4(0.8)=5$ as required.