Half angle formula , what is happening in these formulas

[Jurrasic]

The half angle formulas used when you solve problems that are like the following I mean: Ok, they usually look something like (I may have left something out, but I am putting down the general kind of idea of how they look and work:)
Something like, Find Cos A/2 if Sin A =-1/2 and SinA is in Quadrant 3
Then you have to put Cos A between these things <cosA< and then 180 goes on one side of that and 270 goes on the other side, or what have you, and then you divide 180 and 270 in half, but aren't we just dividing an angle in half, so why does that change what quadrant it's in? That is so weird? lol
So what are these formulas having you do when you use them? Apparently they aren't dividing an actual angle in half. So why do they even call them half angle formulas.

 

[PeterO]

The half angle formulas used when you solve problems that are like the following I mean: Ok, they usually look something like (I may have left something out, but I am putting down the general kind of idea of how they look and work:)
Something like, Find Cos A/2 if Sin A =-1/2 and SinA is in Quadrant 3
Then you have to put Cos A between these things <cosA< and then 180 goes on one side of that and 270 goes on the other side, or what have you, and then you divide 180 and 270 in half, but aren't we just dividing an angle in half, so why does that change what quadrant it's in? That is so weird? lol
So what are these formulas having you do when you use them? Apparently they aren't dividing an actual angle in half. So why do they even call them half angle formulas.

Your example was a little off:

Something like, Find Cos A/2 if Sin A =-1/2 and SinA is in Quadrant 3

should read

Something like, Find Cos A/2 if Sin A =-1/2 and A is in Quadrant 3.

Also, if 180<A<270, then 90 < A/2 < 135 which is in the second quadrant, and we thus know what sign should be in the answer to Cos A/2. That is useful if your calculations present you with two possible answers, and one of them is positive and the other negative.

I believe the half angle formulae are just adaptations of the Double angle formulae - such as Sin2A = 2.SinA.CosA

 

[hubewa]

You should first try to find the value of A based on the information and then find cos A/2. You only need to worry about the limits for sin A.

 

[PeterO]

The half angle formulas used when you solve problems that are like the following I mean: Ok, they usually look something like (I may have left something out, but I am putting down the general kind of idea of how they look and work:)
Something like, Find Cos A/2 if Sin A =-1/2 and SinA is in Quadrant 3
Then you have to put Cos A between these things <cosA< and then 180 goes on one side of that and 270 goes on the other side, or what have you, and then you divide 180 and 270 in half, but aren't we just dividing an angle in half, so why does that change what quadrant it's in? That is so weird? lol
So what are these formulas having you do when you use them? Apparently they aren't dividing an actual angle in half. So why do they even call them half angle formulas.

you could use a half angle formula directly: like CosA = 2 Cos^2(A/2) - 1

By transposing this Cos(A/2) would be the square root of (CosA +1)/2

But is is the positive square root, or the negative square root?

By telling you which quadrant you are working with, you know which root to accept.

 

[HallsofIvy]

The half angle formulas used when you solve problems that are like the following I mean: Ok, they usually look something like (I may have left something out, but I am putting down the general kind of idea of how they look and work:)
Something like, Find Cos A/2 if Sin A =-1/2 and SinA is in Quadrant 3
Then you have to put Cos A between these things <cosA< and then 180 goes on one side of that and 270 goes on the other side, or what have you, and then you divide 180 and 270 in half, but aren't we just dividing an angle in half, so why does that change what quadrant it's in? That is so weird? lol
So what are these formulas having you do when you use them? Apparently they aren't dividing an actual angle in half. So why do they even call them half angle formulas.

I strongly recommend that you learn what formulas mean instead of just memorizing symbols!

"Cos A between these things <cosA< and then 180 goes on one side of that and 270 goes on the other side, or what have you", is mostly nonsense. What you should have is 180< A< 270, not "cos A". That means that the angle, A, lies between 180 degrees and 270 degrees, exactly what is meant by " A is in the third quadrant". If 180< A then, dividing both sides by the positive number 2, 90< A/2. Similarly, if A< 270, then A/2< 135. Putting those together, 90< A/2< 135 so A/2 is in the first quadrant. That tells you that all trig functions of A/2 will have positive values and, as others have told you, that will tell you which sign to use for the square root.