Question

If x = cos 39° + sin 57°, y = cos 40° + sin 58° and z = cos 41° + sin 59°, then:
x > y
y > z
x = y = z
x + z > y

Answer 1

given info : x = cos 39° + sin 57°, y = cos 40° + sin 58° and z = cos 41° + sin 59°

solution : x = cos39° + sin57°

= cos(90° - 51°) + sin57°

= sin51° + sin57°

using formula, sinA + sinB = 2sin(A + B)/2 cos(A - B)/2

= 2sin54° cos3°

similarly, y = cos40° + sin58°

= sin50° + sin58°

= 2sin54° cos4°

z = cos41° + sin59°

= sin49° + sin59°

= 2sin54° cos5°

we see, x = (2sin54°) cos3°

y = (2sin54°) cos4°

z = (2sin54°) cos5°

we also know, cosine is decreasing from 90° to 0°

so, cos3° > cos4° > cos5°

therefore, x > y > z

Therefore the correct options are x > y and y > z