Arcsin, arccos, etc.

[Daimeera]

I'm working on math correspondence courses which unfortunately offer very few resources for help. I just learned about special angles and now am reading about things such as x = {90 degrees +360 degrees k, k ε I} which I'm sure has an official name that they neglected to tell us. I'm sailing along okay with that and then they start talking about arcsines.

There are no explanations on how to use these, simply a question:

"Denise discovered that, when solving the equation in Question 26 (the equation was -1 = 3(cos x)-1) algebraically, she could find the value using the cos^-1 key on her calculator. However, the calculator produces only one answer. How does she know if there are more answers or not? If there are more answers, how does she know what they might be?"

It's not so much the question that's confusing me, it's the use of the arccos at all. I don't understand what to enter into the calculator in order to solve the equation (I'm assuming I'm trying to solve for x).

I think the first part of the answer is simply that all cosine (and sine) equations indicate a repeating pattern of y values, therefore there will always be multiple x-values that correspond with your y answer. Hopefully I'm correct on that. I can even assume that the second response is something related to adding 360 degrees times any integer to the answer.

But I'm still lost as to how to use the arccos, and therefore I'm not even sure if my answer is right . . . but even if it is, that's not helpful if I don't understand everything involved.

Thanks so much!

 

[mark07]

You need to look up the definition of ARCCOS. See, COS is not one-to-one on its domain of real numbers. You need to restrict the domain to be able to define its inverse. There isn't a unique way to do this. However, the standard way of defining ARCCOS is by restricting the domain of COS to [0,pi] where it's one-to-one, and defining y=arccos(x) whenever x=cos(y). With that definition, domain of ARCCOS is [-1,1] and its range is [0,pi].

When you enter ARCCOS(1/2) into the calculator, it will give you pi/3, an answer between 0 and pi. But you know that COS is 1/2 at infinitely many points. So when you solve equations involving COS, and use ARCCOS function, you need to be careful.

Summary: ARCCOS is the inverse of COS restricted to [0,pi].