# Properties of Integration

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!

## Answers and Replies

Simon Bridge
Science Advisor
Homework Helper
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
Staff Emeritus
Science Advisor
Homework Helper
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
Science Advisor
Homework Helper