sqrt(sin^2(x))?

Question

I was wondering is it possible to say that sqrt ( sine squared x) is sine. Or can say (sin^2(x))^1/2=sin(x )is that possible?

Answer 1

Actually, the correct answer is

√(sin^2(x))=|sin(x)|.

Why is it NOT ±sin(x)? There is a simple explanation for this:

Is f(x)=√(sin^2(x)) a function? In other words, for every number that I put in for x, do I get 1 number out? Of course.

Is f(x)=±sin(x) a function? No, it isn't, for example f(π/2)=±1. Therefore √(sin^2(x)) cannot possibly be ±sin(x).

As a college instructor, I have seen people make this mistake all the time. Again, COLLEGE! It is not that easy of a thing for some people to grasp.

So, I have explained why it ISN'T ±sin(x), but I haven't explained why it IS |sin(x)|.

First of all, one of the definitions of |x| is
|x|=√(x^2)

From this, it is clear that my answer is correct, but I don't want to leave you hanging on why |x|=√(x^2), so I will finish with that explanation.

Take any positive number a. Then a^2 is also positive. Now √(a^2) is the non-negative number x such that x^2=a^2. Clearly a works for this. So if a is positive √(a^2)=a.

Now take any negative number -a (so a is positive). Then (-a)^2=a^2. And √(a^2) is again a (since a is positive). Therefore √((-a)^2)=a, or by letting b=-a, we have √(b^2)=-b.

Finally √(0^2)=√0=0.

Thus √(x^2) = x if x≥0, and √(x^2) = -x if x<0. This is the definition of |x|.

Answer 2

Almost, you can say it is +/- sin(x).