0

Link to image: https://i.redd.it/uj2a83j468yy.png
Is this really true? I can't prove or disprove it.

Chris K
  • 63
  • 7
  • 3
    Welcome to Maths SX! This is not only true, it is the rigourous definition of $\sin x$. To prove it in an elementary context, use the power series expansion of $\mathrm e^u$, and set $u=ix$: $\sin x$ is the imaginary part, by Euler's formula. – Bernard May 19 '17 at 00:06
  • 4
    Ultimately it depends on how you're defining the $sin$ function. If you're not already familiar with Taylor series it's not going to be easy, so if you want to learn how to write a formal proof of this I'd recommend you read up on some real analysis notes. In my real analysis lectures, we defined the $sin$ function like this, so it's true by definition. – B. Mehta May 19 '17 at 00:07
  • 1
    The more traditional understanding is that $\sin x$ is defined geometrically, and from that you prove (i) what its derivatives are and then (ii) the formula you gave, which is the Taylor series for $\sin x$. It's in calculus books, e.g., Stewart's Calculus secs. 3.5, 12.10 and Appendix D (1999 ed.). An exemplary free calculus book: http://www.gutenberg.org/ebooks/38769?msg=welcome_stranger – ForgotALot May 19 '17 at 00:41

1 Answers1

1

If you start with $\sin'(x) = \cos(x), \cos'(x) = -\sin(x), \sin(0) = 0, \cos(0) = 1, \sin^2(x)+\cos^2(x) = 1$, you can proceed like this (not original with me):

$$\sin(x) =\int_0^x \cos(t)dt \le\int_0^x dt =x $$ $$\cos(x)-\cos(0) =\int_0^x -\sin(t) dt =-\int_0^x \sin(t) dt \ge-\int_0^x t dt =-\frac{x^2}{2}\\ \text{ so } \cos(x) \ge 1-\frac{x^2}{2} $$ $$\sin(x) =\int_0^x \cos(t)dt \ge\int_0^x (1-\frac{t^2}{2})dt =x-\frac{x^3}{6} $$ $$\cos(x)-\cos(0) =\int_0^x -\sin(t) dt =-\int_0^x \sin(t) dt \ge-\int_0^x (t-\frac{t^3}{6}) dt =-\frac{x^2}{2}+\frac{x^4}{24}\\ \text{ so } \cos(x) \le 1-\frac{x^2}{2}+\frac{x^4}{24} $$

By induction you can derive the power series for sin and cos and show that they are enveloping (the sum is between any two consecutive sums).

marty cohen
  • 107,799