1. ## Inverse Cosine Function

Hi,

Are the general solutions for cos(x) = cos(a), x = 2*pi*k +/- a (where k is an integer)? If so, I was trying to find the general solution to cosx = 0, I thought it would be 2*pi*k +/- pi/2, however there solution was pi/2 + k*pi. How did they get there solution?

Thanks

2. ## Re: Inverse Cosine Function

$\noindent This is a special case. Do you know the Unit Circle? \\ For the ''how?'': it's basically because \cos has roots at x= 0, \pi/2,-\pi,2. From what you know, \cos has periodicity 2\pi. Then since x=\pi/2,-\pi/2 are roots, then x=\pi/2 - 2\pi,-\pi/2 + 2\pi are roots as well. This can keep repeating. It just happens that we can ''condense'' this ''repeating'' solution with 0 + k\pi/2 for k\in\mathbb{Z}.\\ There is a similar case for \cos(x) = 1.$

3. ## Re: Inverse Cosine Function

Oh, ok. Yeah I do, I just didn't think of it like that, I was just substituting it into the formula (I'll have to think more carefully next time). For the cosx=1, is it 2*k*pi from the formula?

4. ## Re: Inverse Cosine Function

Oh yes, I meant there is a similar case for sinx = 0.
(Try and see the cases using the Unit Circle, it helps a lot).

5. ## Re: Inverse Cosine Function

Ok, so I think the simplified version for sinx=0 is x=pi/2 +2kpi, is that correct?

6. ## Re: Inverse Cosine Function

Originally Posted by frog1944
Ok, so I think the simplified version for sinx=0 is x=pi/2 +2kpi, is that correct?
No, for sinx=0, it is x=kpi.
I think you may have mistaken x=pi/2+2kpi as the general solution to sinx=1 (which is correct!)

7. ## Re: Inverse Cosine Function

Originally Posted by frog1944
I was trying to find the general solution to cosx = 0, I thought it would be 2*pi*k +/- pi/2, however there solution was pi/2 + k*pi.
Those two solution sets are the same (can you see why?).

Also, for finding the general solution for cos(x) (or other trig. functions) equal to some "special" values (basically 0 or ±1), it is usually easier to inspect the general solution from the graph rather using the general solution formula (the general solution formula will still give you the correct solution set, but it won't be in as "simplified" a form as it could be).

8. ## Re: Inverse Cosine Function

Originally Posted by fluffchuck
No, for sinx=0, it is x=kpi.
I think you may have mistaken x=pi/2+2kpi as the general solution to sinx=1 (which is correct!)
You're correct fluffchuck, my bad, thanks for pointing it out .

9. ## Re: Inverse Cosine Function

Originally Posted by InteGrand
Those two solution sets are the same (can you see why?).
Yeah I think I can now see why they're the same, by your suggestion of looking at the graph. Awesome! Thank's Integrand, I'll keep in mind to look at the graph when they ask about general solutions to trigonometric functions.

There are currently 1 users browsing this thread. (0 members and 1 guests)

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•