I'm doing math practice problems from "Precalculus for Dummies 1,000 Practice Problems Book" and I'm confused about when to apply restrictions to trig function questions. This book has all the solutions step by step in the back so I know how the problem is solved. What confuses me is why the restrictions are used in some problems and not others.
The restrictions that I'm talking about are:
The restrictions for the regular trig functions should be the the same but the range and domain's values are switched if I'm not mistaken.
These were the main problems that made me confused:
Find an exact value of yy, y=arcsin(√32)
Find an exact value of y, y=cos(arctan(−1))
in this case, while solving for y, I get to a step that looks like this:
y=cos(7π/4). The answers/solutions in the back then show y=√2/2 is the answer.
However, I thought cos(x)'s domain should have been restricted to [−1,1]. Why is the 7π/4 allowed?
At this point I was thinking, then do the restrictions only apply to inverse trig functions? But that didn't make sense either because I should be able to rewrite y=cos(7π/4) as arccos(y)=7π/4; the "7π/4" part is not part of arccos's range restriction, [0,π]
Find all solutions of the equation in the interval [0∘,360∘). 2cos2x−2=3cosx
This confuses me because question 2 allows for cosx's domain to be outside of the restriction while question 3 doesn't allow for cosx's range to be outside of the restriction.
