Is Trigonometry done differently in the us?

Question

Im italian and I've watched some videos from americans and noticed a weird thing. Let's talk about a linear trigonometric equation like this: sinx+cosx+sqrt(3)=0, I've seen americans solving it by just doing an irrational equation, which takes more time and is pointless. Here we are taught that there are 3 ways to solve it: 1)We call sinx=Y and cosx=X and do a system with the equation of the trigonometric circle, x2+y2=1, then we find the values of the sine and cosine and find the related angles, we call this the graphic method. 2) We call sinx=a and cosx=b, calculate r=sqrt(a2+b2) and alfa=tg-1(b/a) and use the fornula r*sin(x+alfa)>c where c in this case is -sqrt(3). We call this method of the auxiliary angle. 3) We use the formulas sinx=2t/1-t2 cosx=1-t2/1+t2 where t is tg(x/2) and solve the equation. Isn't this taught?

Answer 1

Students are often taught to put the equation entirely in either sine or cosine to basically make it into a quadratic equation. It is a longer method but something straight forward. Clever methods can cause panic in some American students, because they feel like that they cannot remember them during a test

Answer 2

Honestly, I'm not sure that they teach this as an important problem. They certainly don't teach this particular problem as it has no solution. Here is how I approached it:

sinx + cos x = -sqrt(3)

(sinx+cosx)^2 = 3

sin²(x)+cos²(x) +2sinxcosx = 3

2sinxcosx = cos(2x) = 2 Not possible.

The max sum of |sin + cos| is sqrt(2) at 45°