Find the complete set of values of x, with 0 ≤ x ≤ π, for which (1 − 2sin x)cos x ≥ 0
- A0 ≤ x ≤ π6, π2 ≤ x ≤ 5π6
- B0 ≤ x ≤ π6, 5π6 ≤ x ≤ π
- Cπ6 ≤ x ≤ π2, 5π6 ≤ x ≤ π
- Dπ6 ≤ x ≤ 5π6
Show the answer and worked solution
answer · A
- A0 ≤ x ≤ π6, π2 ≤ x ≤ 5π6
- B0 ≤ x ≤ π6, 5π6 ≤ x ≤ π
- Cπ6 ≤ x ≤ π2, 5π6 ≤ x ≤ π
- Dπ6 ≤ x ≤ 5π6
A product is non-negative when both factors have the same sign, so treat the two factors separately on [0, π]. Here 1 − 2sin x ≥ 0 means sin x ≤ 12, which holds on [0, π6] and on [5π6, π]; and cos x ≥ 0 holds on [0, π2]. Both non-negative gives 0 ≤ x ≤ π6. Both non-positive needs π6 ≤ x ≤ 5π6 together with π2 ≤ x ≤ π, that is π2 ≤ x ≤ 5π6.