Whether root should be positive or negative

In summary, the conversation discusses finding the expression for ##\sin (\arccos x)## and determining whether to use the positive or negative root. It is determined that the sign depends on the sign of x and that both signs may be correct for different values of x. The range of the inverse trigonometric functions is also discussed in relation to determining the sign.
  • #1
1,462
44
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...
 
Mathematics news on Phys.org
  • #2
##0 \leq \arccos(x) \leq \pi##, which means the sine of it is always positive.
 
  • Like
Likes Mr Davis 97
  • #3
mfb said:
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?
 
  • #4
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.
 
  • #5
mfb said:
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...
 
  • #6
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.
 
Last edited:
  • #7
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.
 
  • Like
Likes Mr Davis 97

1. What is the significance of a positive or negative root in a scientific context?

A positive root indicates a solution or value that is greater than zero, while a negative root indicates a solution or value that is less than zero. This can have different interpretations depending on the specific context, but it is often used in equations or models to represent different outcomes or scenarios.

2. How do you determine whether a root should be positive or negative?

The determination of a positive or negative root depends on the specific equation or problem being solved. This can be done through various methods such as graphing, substitution, or using mathematical principles like the quadratic formula. It is important to carefully consider the context and meaning of the root in order to determine the correct sign.

3. Can a root be both positive and negative?

In some cases, a root can have both positive and negative solutions. For example, in a quadratic equation, there can be two solutions, one positive and one negative. This is often represented as ± (plus or minus) to indicate both possibilities. However, in other cases, such as finding the square root of a number, there is only one possible solution, so the root cannot be both positive and negative.

4. How does a positive or negative root affect the overall outcome of an experiment or study?

The effect of a positive or negative root on the outcome of an experiment or study depends on the specific context and variables involved. In some cases, a positive root may indicate a desired outcome, while in others, a negative root may be more desirable. It is important for scientists to carefully consider the implications of both positive and negative roots in their research.

5. Can a root ever be neither positive nor negative?

In mathematics, roots are typically either positive or negative. However, in certain contexts, a root can be considered neither positive nor negative. For example, in complex numbers, there are solutions that are neither strictly positive nor negative. It is important to consider the specific context and mathematical principles when determining the nature of a root.

Similar threads

Replies
13
Views
3K
Replies
2
Views
999
Replies
2
Views
1K
  • General Math
Replies
4
Views
916
Replies
4
Views
907
  • General Math
2
Replies
45
Views
3K
  • General Math
Replies
1
Views
7K
  • General Math
Replies
5
Views
896
Replies
3
Views
853
  • General Math
Replies
3
Views
1K
Back
Top