2cosθ=⅓ or √3 or 1/√3 or 1.3? The meaning is not clear, so I’ll replace the right hand side with a, therefore cosθ=a/2. In a right triangle in which one leg has length a and the hypotenuse is 2. So the other leg has length √(4-a²). From this we know that 4-a²>0, so -2<a<2 and sinθ=½√(4-a²). 2sinθ+tanθ=sinθ(2+1/cosθ)=√(4-a²)(1+1/a).
But we are told that tanθ<0 so sinθ<0, making sinθ=-½√(4-a²).
So 2sinθ+tanθ=-√(4-a²)(1+1/a).
We can now calculate the the expression for each possible value of a so far identified.
a=⅓: -4√35/3
a=√3: -(1+1/√3)=-(1+√3/3)
a=1/√3: -(√33/3+√11)
a=1.3: -(√231/10)(22/13)=-11√231/65
The irrationals are left without calculating since no calculator is to be used to get the solution.