Math Is Fun Forum

Well, for a start, you could teach your program

(1) not to move knights to the sides of the board unnecessarily (knights are more useful in the centre of the board, where they are more mobile)

(2) not to touch the a- and h-pawns until later in the game (the side pawns early on are better engaged in defending the more central pawns, which should be deployed to take control of the centre of the board as quickly as possible)

Do you mean you want angles rotated clockwise to be negative and those rotated counterclockwise to be positive? Then try this formula:

This is computed using the cross product rather than the dot product. In your example (following your style in taking the downward y-direction as positive):

And the cross-product formula gives

While

and the angle with the first vector is

Welcome to the forum, mrfandus.

I know bobbym has kindly helped you out here, but this is your second time here now. Are you actually putting any of your own effort into doing your own homework, or just waiting for nice people like bobbym to give you something for you to copy?

From my experience, Spanish is easy to learn. It is also very widely spoken, so it could come in handy. Therefore, Spanish would be my recommendation.

There is no necessity to prove the converse, which is merely establishing that the inverse function is well defined. Still, if you want to prove it, use the fact that

To prove ontoness, take an

Yes, it is. You just need to get used to the trick.

Equate imaginary parts to solve for each.

All of them.

I posted my own little rant at the other thread.

You’re welcome. I just realized however that you posted this as a real-analysis problem, so it may be that you have to use calculus here. While Jensen’s inequality gives a quick solution without using calculus, it may not be what your teacher wants. Do check with your teacher on what is required for this problem.