Finding Minimum Value

Question

The question in the textbook is:

if a,b,c are unit vectors, then $|a-b|^2+|b-c|^2+|c-a|^2$ does not exceed 'x'. Find x, where x is an integer.

On solving we get this expression: $6-2(a.b + b.c + c.a)$

Now, I understand that we can use $|a+b+c|^2 >= 0$ to find the minimum value of the $(a.b + b.c + c.a)$ term which would give us the value for 'x'.

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But I was wondering wether it would be possible to find the minimum value of the expanded version of the $(a.b + b.c + c.a)$ term?

$(a.b + b.c + c.a) = cosθ$₁ + cosθ₂ + cosθ₃

Here's what I think, in a normal case where θ₁, θ₂ and θ₃ would be independent, the minimum value would be -3 but the angles θ₁, θ₂ and θ₃ are not really independent but bounded geometrically.

So, is it possible to find the minimum value of cosθ₁ + cosθ₂ + cosθ₃?

EDIT: I'm in High School, forgot to mention that.