For 0 <= x <= 2*pi, sin(2x) and cos(x) are both decreasing functions in the interval _________. A. (0, pi/2) B. (pi/4, 3*pi/4) C. (pi/2, pi) D. (3*pi/4, 2*pi)
GATE 2018 · General Aptitude · Trigonometry · medium
Answer: Both functions are decreasing on (pi/4, 3*pi/4).
- Find where sin(2x) is decreasing: 2*cos(2x) < 0 => cos(2x) < 0. cos is negative in (pi/2, 3*pi/2), so pi/2 < 2x < 3*pi/2 => pi/4 < x < 3*pi/4.
- Find where cos(x) is decreasing: -sin(x) < 0 => sin(x) > 0. sin is positive on (0, pi), so cos(x) is decreasing on (0, pi).
- Intersect the two intervals: (pi/4, 3*pi/4) is fully contained within (0, pi), so the intersection is (pi/4, 3*pi/4).