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A question is given in my book which I'm unable to solve.

If $\displaystyle\sum_{i=1}^{2022} \sin^{-1}(x_i) = 1011\pi $, then what is the value of $\displaystyle\sum_{i=1}^{2022} x_i$?

Answer of the above problem is given to be $2022$.

In general, this problem is of the form that if $\displaystyle\sum_{i=1}^{n} A_i = k$, then what is the value of $\displaystyle\sum_{i=1}^{n} \sin(A_i).$ So we need to find the sum of sines of numbers, when sum of numbers is given. But I can't continue from here because of my little knowledge.

Alternatively, I thought of using the formula:
$\sin^{-1}(x) + \sin^{-1}(y) = \sin^{-1}(x\sqrt{1-y^2} + y \sqrt{1-x^2})$ to simplify the sum $\displaystyle \sum_{i=1}^{2022} \sin(x_i)$. But here, because of $2022$ terms, it's very difficult to apply the formula.

I think there would be surely any method/formula for simplifying $\sum_{i=1}^n \sin^{-1}(x_i)$ in general, which I'm unaware of.

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    $\begingroup$ Try this: If $\displaystyle\sum_{i=1}^{2} \sin^{-1}(x_i) = 2$, then what is the value of $\displaystyle\sum_{i=1}^{2} x_i$? $\endgroup$
    – Will Jagy
    Commented Jan 16, 2023 at 16:50
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    $\begingroup$ Oops! There is a typo in question. Just edited. Sorry. $\endgroup$
    – Utkarsh
    Commented Jan 16, 2023 at 16:54
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    $\begingroup$ Now that the question is fixed: think about the range of $\arcsin$. What is the largest and smallest value this function can attain? So what is the largest and smallest value of the sum, and what inputs are needed to attain this sum? $\endgroup$
    – Steven Gubkin
    Commented Jan 16, 2023 at 16:56
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    $\begingroup$ What's the maximum of $\sin^{-1}(x)$? What's the maximum of $\sum_{i=1}^{2022} \sin^{-1}(x_i)$? $\endgroup$
    – Ennar
    Commented Jan 16, 2023 at 16:57
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    $\begingroup$ Aaaha... Got it. All the inverse sines are at there maximum in the summation. All the $x_i$'s are equal to 1. So that all the $\sin^{-1}(x_i)$ are $\pi/2$. Thanks to all of you! :D $\endgroup$
    – Utkarsh
    Commented Jan 16, 2023 at 16:58

1 Answer 1

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The range of $\sin^{-1}(x)$, where we assume this is the principal value, is $[-\pi/2,\pi/2]$. The only way for the first sum to equal $1011\pi$ is if all summands have the maximum value of $\pi/2$, which is achieved with $x_i = 1$. Therefore, $\sum_{i=1}^{2022}x_i = 2022$.

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