For every real θ, the value of 3co$s^2$θ + 4si$n^2$θ lies between:

Split the second term: 3co$s^2$θ + 4si$n^2$θ = 3(co$s^2$θ + si$n^2$θ) + si$n^2$θ = 3 + si$n^2$θ. Since si$n^2$θ runs from 0 to 1, the expression runs from 3 to 4, and both ends are actually reached, at θ = 0° and θ = 90°. Treating the two coefficients as the extremes suggests 3 and 7 or 0 and 7, while −1 to 1 confuses the whole expression with a plain sine.