Exercise 2: Why a=3 Produces No Valid Triangle at All — Possible Solution
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WHAT b*sin(A) GEOMETRICALLY REPRESENTS
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Picture the triangle being constructed from angle A at one vertex,
with side b drawn out from that vertex at the given angle, and side a
required to swing from the far end of b back around to close the
triangle. The quantity b*sin(A) is the PERPENDICULAR (shortest
possible) distance from the far end of side b down to the line side a
would need to lie along - it represents the minimum possible length
side a could have and still be able to reach back and close the
triangle at all. Side a swinging like a hinge from the endpoint of b
can only close the triangle if it's at least this long; any shorter,
and it simply cannot span the gap.

WHY a=3 SPECIFICALLY FAILS
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This chapter verified that with b=10 and A=40 degrees, b*sin(A)/a
came out to about 2.14 when a=3 - a value greater than 1. Since the
Law of Sines requires sin(B) = b*sin(A)/a, and no real angle B can
ever have a sine greater than 1 (sine is bounded between -1 and 1 for
any real angle), this immediately signals that no valid angle B
exists that would satisfy the given measurements. Geometrically, this
matches the perpendicular-distance explanation directly: b*sin(A)
(the minimum length side a would need) works out to about 6.43 for
b=10 and A=40 degrees, and a=3 is shorter than that minimum - side a
is simply too short to physically reach around and close the
triangle, no matter what angle it's given.

WHY THIS IS A CLEAN, PRINCIPLED EXPLANATION RATHER THAN JUST "THE MATH BREAKS"
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The failure isn't an arbitrary quirk of the formula - it directly
reflects a real geometric impossibility. Checking whether b*sin(A)/a
exceeds 1 is mathematically equivalent to checking whether side a is
at least as long as the minimum perpendicular distance b*sin(A)
requires, which is exactly why this specific check (rather than some
other arbitrary threshold) is the right one to perform before
attempting to solve an SSA triangle.

WHY THIS WORKS AS AN ANSWER
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The explanation connects the algebraic failure condition
(b*sin(A)/a > 1, therefore no valid sine) to a genuine geometric
meaning (b*sin(A) as the minimum reachable length for side a), rather
than treating the domain-error condition as an arbitrary rule to
memorize, and applies it correctly to this chapter's own verified
a=3 case.