How can I get solutions of this trigonometric equation?

Question

I am trying to solve the equation $$\left(\sin (x)+\cos (x)-\sqrt{2}\right)\cdot \sqrt{-11 x-x^2-30}=0$$ in Real domain. I tried

First way

Solve[{(Sin[x] + Cos[x] - Sqrt[2]) Sqrt[-11 x - x^2 - 30] == 0}]

I got

{{x -> -6}, {x -> -5}, {x -> [Pi]/4}}

Second way

   Solve[{(Sin[x] + Cos[x] - Sqrt[2]) Sqrt[-11 x - x^2 - 30] == 0}, x, 
  Reals] // FullSimplify

I got

{{x -> -6}, {x -> -5}, {x -> -((7 [Pi])/4)}, {x -> -((7 [Pi])/4)}}

Third way

sol = (TrigExpand@
  Reduce[(Sin[x] + Cos[x] - Sqrt[2]) Sqrt[-11 x - x^2 - 30] == 0, 
   x, Reals] // FullSimplify // Last) /. C[1] -> k

I got

x == -((7 [Pi])/4)

How to get the correct solutions?

Answer 1

Solve gives you a warning that it may not find all possible answers. Use Reduce instead:

Reduce[{(Sin[x] + Cos[x] - Sqrt[2]) Sqrt[-11 x - x^2 - 30] == 0}]

(C[1] \[Element] Integers && x == -2 ArcTan[1 - Sqrt[2]] + 2 \[Pi] C[1]) || x == -6 || x == -5

which shows that there is an infinite family of solutions, one for each integer C[1].

Answer 2

If you restrict the solution range to -6<= x<= -5 (from checking Sqrt[-11 x - x^2 - 30] )

Solve[{(Sin[x] + Cos[x] - Sqrt[2]) Sqrt[-11 x - x^2 - 30] == 0 // TrigToExp, -6 <= x <= -5}, x]
(* {{x -> -6}, {x -> -5}, {x -> -((7 \[Pi])/4)}, {x -> -((7 \[Pi])/4)}} *)

Solve( not NSolve!!) gives all real solutions