Can Trig Substitution with Cosine be Used Instead of Sine?

[UMath1]

I was wondering if you could do a trig substitution with cosine instead of sine. All the textbooks I have referred to use a sine substitution and leave no mention as to why cosine substitution was not used. It seemed that it should work just the same, until I tried it for the following Fint [sqrt(9-x^2)]/ [x^2]. I checked to see if my answer differed by only a constant but that was not the case. I have attached pictures of my work. Can anyone tell me why it does not work?

 

[blue_leaf77]

If you use sine instead, you will end up with $\sin^{-1}$ in place of $\cos^{-1}$. But the two expressions are related by $\cos^{-1}x = \pi/2-\sin^{-1}x$.

 

[UMath1]

I know but why are the answers different? Is one less valid than the other?

Btw the textbook which uses sine has the same answer but with -sin^-1(x/3) instead of cos^-1(x/3) like I have it.

 

[blue_leaf77]

Btw the textbook which uses sine has the same answer but with -sin^-1(x/3) instead of cos^-1(x/3) like I have it.

That's exactly the point I addressed in my previous post. Replace -sin^-1(x/3) with the equation I wrote before. You will indeed have an additional $\pi/2$ but it's a constant and hence can be absorbed into the integration constant $C$.

 

[QuantumQuest]

There's really nothing magic about using sin or cos. It just depends on what is more convenient for each case. As for signs, using the relevant relations from trigonometry - like the one that blue_leaf77 mentions, you can substitute sin for cos and vice versa and find the appropriate sign.

 

[zinq]

The function being integrated is

f(x) = √(9 - x²) / x²​

. This is defined for 0 < |x| ≤ 3.

When making a substitution we want to choose an interval where f(x) makes sense, and the easiest one is 0 < x ≤ 3.

We also want to choose a substitution that takes the same values that f(x) does over the interval of definition, and that's between 0 and 3.

Each of y = 3 sin(x) and y = 3 cos(x) satisfy this condition, so either one can be used for the substitution.

Using 3 sin(x) to substitute might be a tiny bit easier than cosine because its derivative is 3 cos(x), and this does not introduce negative signs.

 

[UMath1]

Ok...I see it now. I tried some test bounds of integration and got the same answer from both options.