Exercise 3: The Risk of Always Taking the First asin Solution — Possible Solution
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WHAT COULD GO WRONG
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This chapter verified directly that with b=10 and A=40 degrees, a=7
produces exactly two valid, geometrically real triangles: one with
B approximately 66.67 degrees (giving side c approximately 10.43) and
another with B approximately 113.33 degrees (giving side c
approximately 4.89). A positioning system that always takes the first
(asin) solution without checking for a second would silently commit
to the B approximately 66.67 degree answer every time this exact
input pattern occurs, even in cases where the device's true position
actually corresponds to the OTHER valid triangle - producing a
position estimate that satisfies all the given measurements just as
validly, but places the device somewhere genuinely different from
where it actually is, with no error, warning, or indication that a
second possibility even existed.

UNDER WHAT CIRCUMSTANCES THE BUG WOULD ACTUALLY MANIFEST
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The bug only manifests specifically when the input measurements fall
into the "ambiguous case" range this chapter identified - specifically
when the given side a is shorter than side b but longer than the
minimum reachable distance b*sin(A) (from Exercise 2's own reasoning).
Outside that range - when a is too short for any solution, or long
enough that only one solution exists (this chapter's own a=15 case) -
taking the first asin result happens to be harmless, either because
no valid answer exists at all or because there genuinely is only one.
This is exactly what makes the bug dangerous in practice: the system
would work correctly in testing and in the majority of real
measurements, and only produce a wrong position in the specific,
possibly infrequent geometric configurations that fall into the
ambiguous range - making it a classic intermittent, hard-to-reproduce
bug rather than one that fails consistently and gets caught early.

WHY THIS MATTERS MORE FOR A POSITIONING SYSTEM SPECIFICALLY
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Unlike some contexts where picking "a" valid answer among several
might be acceptable, a positioning system's entire purpose is
reporting the device's one true location - silently reporting a
mathematically valid but factually wrong position is arguably worse
than an obvious failure, since nothing in the system's own output
would indicate anything went wrong.

WHY THIS WORKS AS AN ANSWER
------------------------------
The explanation identifies the precise failure mode (silently picking
one of two equally valid solutions with no way to tell which is
correct) using this chapter's own concrete a=7 numbers, and correctly
scopes when the bug would and wouldn't actually appear, framing it as
an intermittent rather than a consistently-reproducible failure.