Whether root should be positive or negative

[Mr Davis 97]

I'm trying to find $\sin (\arccos x)$. I let $\theta = \arccos x$ and then use $\sin ^2 \theta + \cos ^2 \theta = 1$, I get $\sin (\arccos x) = \pm \sqrt{1 - x^2}$. I'm not sure whether to take the positive or negative root. On Wolfram Alpha is shows that the result is the positive root, but I'm not sure why...

 

[mfb]

$0 \leq \arccos(x) \leq \pi$, which means the sine of it is always positive.

 

[Mr Davis 97]

For x>0 you have arccos(x)<pi/2, which means the sine is positive. For x<0 you have arccos(x)>pi/2, which means the sine is negative.
Pick the sign depending on the sign of x.

What if I want a general expression and don't know the sign of x? For example, I am trying to find the antiderivative of $\arcsin x$, and I found that $\int \arcsin x dx = x \arcsin x + \cos (\arcsin x) + c$. So can I not simplify this further, since I don't know the sign of $x$ beforehand?

 

[mfb]

Forget that old post, I thought about cos instead of sin. I fixed it.

If you need a range where the sign changes then treat the cases separately. Or see where the expression comes from, sometimes the inverse functions are not even what you actually want.

 

[Mr Davis 97]

Forget that old post, I thought about cos instead of sin. I fixed it.

If you need a range where the sign changes then treat the cases separately. Or see where the expression comes from, sometimes the inverse functions are not even what you actually want.

I'm still confused... On various tables of integrals I see that $\int \arcsin x dx = x \arcsin x + \sqrt{1-x^2}+ c$, which means that $\cos (\arcsin x) = \sqrt{1-x^2}$. I don't see why they're choosing the positive root over the negative root...

 

[FactChecker]

There is not reliable rule of which sign to use. You must always consider both and only rule one out when there is a step where it does not fit with other known facts. You may have to carry both along a long way and sometimes to the end. In that case, you need to check both answers in the original problem. Both may be correct. In fact, one sign may be correct for some values of x and the other may be correct for other values of x.

In the example of your post, arccos is formally defined to range from $\pi$ to 0 as x goes from -1 to 1. For those inputs, sin() is always positive. That rules out the final answer of the negative root for every value of x.

 

[mfb]

The arcsine goes from -pi/2 to +pi/2, in this range the cosine is positive. Just check the ranges to see which sign is necessary.