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I consider this function $f(x):=x\cos(x)$ for $ x\in I=\left[-\dfrac{\pi}{2},0\right]$.

Can we write this function on this interval with a different expression?

For example, just as a polynome of $\sin(x)$ and $\cos(x)$.

Aryaman Maithani
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Bernstein
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    No${}{}{}{}{}{}$ – TonyK Jul 25 '20 at 11:31
  • Your question is too vague. What is a "different expression"? You can replace $\cos x$ by $\sqrt{1-\sin^2 x}$, does that count? You can probably find more complicated expressions using other trig identities. You can replace $x$ by $-\sqrt{x}^2$, how about that? You can use an infinite series (Taylor series for the cosine), would that count? But if you want to restrict it to polynomials in terms of purely sine and cosine of $x$ multiplied by a polynomial of $x$, what you got is likely as simple as it gets. – Deepak Jul 25 '20 at 11:45
  • @Deepak thank you, what I want is to drop the$ x$ , and obtain this expression just by sinus and cosinus – Bernstein Jul 25 '20 at 11:49
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    Are you trying to get a Fourier Series? – Benjamin Wang Jul 25 '20 at 11:56
  • @Bernstein No, you cannot do that. – Deepak Jul 25 '20 at 11:56
  • Fourier series don't work here because we don't have the periodicity – Bernstein Jul 25 '20 at 12:09
  • Why do you want to write it as a polynomial? (by the way, you can't). What would you be able to do with a polynomial in sine and cosine that you can't do with $x\cos x$? – Gerry Myerson Jul 25 '20 at 12:15
  • to obtain an explicit expression of the inverse in a subinterval on $I$ – Bernstein Jul 25 '20 at 12:46
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    The answer from @Claude is nice, but if you use it to get an explicit (approximate) expression for the inverse function, you'll have to solve a cubic equation. This can be done, but the formula is not simple. The Lambert $W$-function (https://en.wikipedia.org/wiki/Lambert_W_function) is the inverse function to $f(x)=xe^x$, and seeing as how $\cos x=(e^{ix}+e^{-ix})/2$ maybe it can be put to use for inverting $x\cos x$. – Gerry Myerson Jul 26 '20 at 03:36

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In terms of trigonometric functions, you cannot.

However, in terms of $x$, you could have something which is quite nice using a $1,400$ years old approximation $$\cos(x) \simeq\frac{\pi ^2-4x^2}{\pi ^2+x^2}\qquad (-\frac \pi 2 \leq x\leq\frac \pi 2)$$ which makes $$x \cos(x) \simeq\frac{\pi ^2-4x^2}{\pi ^2+x^2}x$$ which, in the interval shows a maximum error of $0.002$.

Claude Leibovici
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  • Thanks, @Claude Leibovic, la Meilleur :). could you give me any reference and also what about an approximation of the $\sin$ – Bernstein Jul 25 '20 at 15:15
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    @Bernstein. Have a look at https://math.stackexchange.com/questions/976462/a-1-400-years-old-approximation-to-the-sine-function-by-mahabhaskariya-of-bhaska – Claude Leibovici Jul 25 '20 at 15:19