How to calculate inverse cosine of two variables

[Wenlong]

Hi, all

I am looking into inverse cosine operations. I have a question like follows:

Let x and y be two variables of degrees, how to separate equation arccos(x+y) into an equation that contains x and y separately? Such as arccos(x+y) = f1(...x) + f2(...y)?

Thank you very much for your consideration. I'll be extremely appreciated if any help.

Regards
Wenlong

 

[BobG]

Are you asking about the sum and difference formula for cosines and sines? If so, you can look that up on the internet in about 10 seconds.

Or are you asking how to prove the formulas, which is a bit harder.

However, there is a cute way of proving it using the algebraic laws for combining powers, x^a x^b = x^(a+b), and Euler's formula, cos x + i sin x = e^(ix). This gets you the sum formulas for both cosine and sine. (hint: use Euler's formula first, which can then be broken back out using the laws for combining powers).

 

[haruspex]

There's no way it could be expressed as a function of x plus a function of y. You can see that by considering partial derivatives.