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Sine of Squared (Posted on 2018-11-15) Difficulty: 3 of 5
The smallest positive solution to sin x = sin x2 is x = 1.

What is the second smallest positive solution?

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

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Solution Analytic solution aided by graphing Comment 5 of 5 |
I realized after graphing sin(x) and sin(x^2) that the first curve is still "on the way up" to its first peak, but the second one has already passed it's first peak and is coming down.  If you draw a horizontal line through the intersection point, then that horizontal line will intersect y=sin(x) somewhere to the right. 
By symmetry, those 2 points are equidistant from x=pi/2, which is the x value where y=sin(x) peaks.
And the two x values of the two intersections of the horizontal line are x and x^2.
So the average value of x and x^2 is pi/2.
So:  x^2 + x - pi = 0

so x =
[-1 + sqrt(1 + 4 pi)]/2 = 1.341627718...
and
[-1 - sqrt(1 + 4 pi)]/2 = -2.341627718...

Answer to our question:  [-1 + sqrt(1 + 4 pi)]/2

---
edit:  Jer beat me to it, but we got there slightly different ways

Edited on November 16, 2018, 2:02 pm
  Posted by Larry on 2018-11-16 13:59:14

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