Plus/minus in inverse trig? (1 Viewer)

[sinophile]

Ive scanned a question which I was confused about. Where ive circled it, couldn't you consider both negative and positive cases, thus giving two solutions? Can someone explain why you only consider the positive case? Or anything else?

Apologies for it being so big.

 

[maths-bbqz]

Heres my thought:
when it's the square root of x squared, you only take the positive sol'n (since a square root must be positive) but if you have x squared = 1 to start off with, you take both the positive and the negative answer
for example, root 5 doesn't equal minus root root 5, since how can a square root of something produce a negative result over the real field? But if x squared equals 5, you would take both solutions
Hope this helps and sorry i dont know how to use the equation editor D:

 

[Trebla]

For the inverse cosine function to exist, it can only take a derivative of - 6. Its domain would be:



in order for it to be a function. Under this domain, cos 2x is strictly positive so we don't worry about the negative case.

However, under other circumstances, you would need to consider both cases...