Properties of Integration

[MathewsMD]

Hi,

Recently in class, my professor went over a relationship that exists between trigonometric functions, T(x), and their complementary functions. That is:

∫ T(x)dx = W(x) + C
∫ coT(x)dx = -co[W(x)] + C

Without really providing a proof, we were told this relationship. I've plugged in numbers and tried graphing the scenario and it works from what I've done so far. If anyone could provide me a formal proof through any source, that would be great since I need a bit more convincing.

Also, is a limitation to this property that both W(x) and T(x) must be a combination trigonometric functions themselves? Since when trying to solve ∫cscxdx using the answer from ∫secdx, an answer cannot be found. Are there any other limitations?

Also, regarding the identities:
sinAcosB = 1/2 [sin(A-B) + sin(A+B)
sinAsinB = 1/2 [cos(A-B) - cos(A+B)]
cosAcosB = 1/2[cos(A-B) + cos(A+B)]

Is there a formal proof readily accessible for these identities?

Thanks!

 

[Simon Bridge]

Comes from the definitions of the trigonometric functions on the unit circle... which tells you how they are related to each other.

For the identities:
Helps to noticing that for a point (x,y) in the complex plane, the vector from the origin to (x,y) makes an angle $\theta$ to the real axis:

$e^{i\theta}=\cos\theta+i\sin\theta = x+iy$

You can use that for the integral of trig things too, since $\int e^x dx = e^x +c$

 

[SteamKing]

Also, regarding the identities:
sinAcosB = 1/2 [sin(A-B) + sin(A+B)
sinAsinB = 1/2 [cos(A-B) - cos(A+B)]
cosAcosB = 1/2[cos(A-B) + cos(A+B)]

Is there a formal proof readily accessible for these identities?

Thanks!

Yes, all it takes is a little manipulation of the basic identities for the sine and cosine of the sum and difference of two angles:

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

 

[Simon Bridge]

... which turns the question into "is there a formal proof of the sine and cosine of the sum and difference identities?" :)
http://www.themathpage.com/atrig/sum-proof.htm