Question

If x=sin 14 θ+cos 20 θ,θbelongs of R,then range of x is?

Answer 1

Range of x  ≈  [-2,  1.96697]

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x = sin 14A  +  cos 20A ,        where A ∈ R.
It is enough to consider 0<=A<=π, as x has a periodicity of π. can verify that by finding x at  (A+π).

x = Sin [14(A+π/7)]  + Cos [20(A+π/10)]
 
-1 <= Sin 14A <= 1    and   -1 <= Cos 20A  <= 1 

  so the Maximum value of x is <= 2   and  minimum value >= -2.  We need to find the exact max and min values...

For A = π/4,   x = sin 7π/2 + cos 5π = -2   minimum value..

For A = 50π/83 and 749π/830,  x ≈ 1.96697

Maximum value is just less than 2.
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This can be found using derivatives..

dx/dA = 14 cos 14A - 20 sin20A   = 0  
         => sin20A = 0.7 Cos 14A         at the extremum
   so Cos 20A = + √[1 - 0.49 Cos² 14A] = + √[0.51 + 0.49 sin² 14A]

max or min x = Sin 14A + √[0.51 + 0.49 Sin² 14A]
     when  sin14A ≈1,  then x ≈ 2 ..
               sin 14A ≈ -1,  then x ≈-2..