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Math Help - Please provide proof for the main trig functions

  1. #1
    Senior Member Paze's Avatar
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    Please provide proof for the main trig functions

    Hi MHF. If anyone has time, I would love for concrete proof for the following trig functions:

    sin(u+v)=sin(u)\cdot cos(v)+cos(u)\cdot sin(v)\\sin(u-v)=sin(u)\cdot cos(v)-cos(u)\cdot sin(v)\\cos(u+v)=cos(u)\cdot cos(v)-sin(u)\cdot sin(v)\\cos(u-v)=cos(u)\cdot cos(v)+sin(u)\cdot sin(v)\\sin(2u)=2\cdot sin(u)\cdot cos(u)\\cos(2u)=cos^2(u)-sin^2(u)\\sin^2(u)+cos^2(u)=1

    I do know the last one but I thought I'd have it here to have the whole bulk for search engines etc. in case of a good answer on the thread.

    Thanks a lot!
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    Newbie Esteban's Avatar
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    Re: Please provide proof for the main trig functions

    Those aren't simple proofs...

    here you have a link of a math forum from my country with those proofs (it's in spanish :S, but you can understand if you only read the operations)

    Basically, getting 1st and the 3rd and using parity and imparity of cos and sin you can demostrate the others

    Funciones trigonométricas de la suma y diferencia de ángulos - Foro fmat.cl

    I hope it can help you
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  3. #3
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    Re: Please provide proof for the main trig functions

    Quote Originally Posted by Paze View Post
    Hi MHF. If anyone has time, I would love for concrete proof for the following trig functions:

    sin(u+v)=sin(u)\cdot cos(v)+cos(u)\cdot sin(v)\\sin(u-v)=sin(u)\cdot cos(v)-cos(u)\cdot sin(v)\\cos(u+v)=cos(u)\cdot cos(v)-sin(u)\cdot sin(v)\\cos(u-v)=cos(u)\cdot cos(v)+sin(u)\cdot sin(v)\\sin(2u)=2\cdot sin(u)\cdot cos(u)\\cos(2u)=cos^2(u)-sin^2(u)\\sin^2(u)+cos^2(u)=1

    I do know the last one but I thought I'd have it here to have the whole bulk for search engines etc. in case of a good answer on the thread.

    Thanks a lot!
    2. Sin, Cos and Tan of Sum and Difference of Two Angles

    As for the double angle formulas, it's simply a case of letting \displaystyle \begin{align*} \beta = \alpha \end{align*} and simplifying.
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    Re: Please provide proof for the main trig functions

    Please provide proof for the main trig functions-geometric-proof.png
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    Re: Please provide proof for the main trig functions

    Please provide proof for the main trig functions-cos-u-v.png
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    Re: Please provide proof for the main trig functions

    How you prove those things depends upon how you define sine and cosine.

    The simplest proofs I know start from the definitions
    "y= cos(x) is the function satisfying the differential equation y''= -y with initial conditions y(0)= 1, y'(0)= 0"
    "y= sin(x) is the function satisfying the differential equation y''= -y with initial conditions y(0)= 0, y'(0)= 1"

    It is easy to prove that sin(x) and cos(x) are independent functions so that any solution to y''= y' can be written as a linear combination of them. In fact, the solution to y''= -y, with initial conditions y(0)= A, y'(0)= B, is y(x)= Acos(x)+ Bsin(x).

    If we define y(x)= (sin(x))', then y'= (sin(x))''= - sin(x) so that y''= (- sin(x))'= -y while y(0)= 1 and y'(0)= 0 so that (sin(x))'= cos(x). If we define y(x)= (cos(x))', then y'= (cos(x))''= -cos(x) so that y''= (- cos(x))'= -y while y(0)= 0 and y'(0)= -1 so that (cos(x))'= -sin(x).

    If we define y(x)= sin^2(x)+ cos^2(x), we have y'= 2sin(x)cos(x)- 2cos(x)sin(x)= 0 for all x. That is, y(x) is a constant and y(0)= (sin(0))^2+ (cos(0))^2= 0^2+ 1^2= 1 so sin^2(x)+ cos^2(x)= 1 for all x.

    Note that y(x)= cos(x+a) satisfies the differential equation y''= -y with initial conditions y(0)= cos(a), y'(0)= -sin(a) so that y(x)= cos(x+ a)= cos(a)cos(x)- sin(a)sin(x). Setting x= b, cos(a+ b)= cos(a)cos(b)- sin(a)sin(b).

    Similarly, y(x)= sin(x+ a) satisfies the differential equation y''= -y with initial conditions y(0)= sin(a), y'(0)= cos(a) so that y(x)= sin(x+a)= sin(a)cos(x)+ cos(a)sin(x). Setting x= b, cos(a+ b)= sin(a)cos(b)+ cos(a)sin(b).

    You can also define sin(x)= \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}, cos(x)= \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}, prove from that that sin(x) and cos(x) satisfy y''= -y with the above initial conditions and then use this proof.
    Last edited by HallsofIvy; March 1st 2013 at 06:52 AM.
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