If sinθ + cosθ = 5/4, then si$n^3$θ + co$s^3$θ equals:

Squaring gives 1 + 2sinθ·cosθ = 25/16, so sinθ·cosθ = 9/32. Then si$n^3$θ + co$s^3$θ = (sinθ + cosθ)(si$n^2$θ − sinθ·cosθ + co$s^2$θ) = $(5/4)(1-9/32)=(5/4)(23/32)=115/128$ ≈ 0.898. Cubing the given number gives 125/64; using $a^2+ab+b^2$ in the bracket gives 205/128; and dropping the outer factor in the form $s^3-3ps$ gives 71/64.