Answer:
4) a > b·sin(A) -- triangle exists
5) a > b·sin(A) -- triangle exists
Explanation:
The triangle will exist when the missing angle value can be computed using the Law of Sines. That tells you ...
sin(B)/b = sin(A)/a
sin(B) = b·sin(A)/a
In order for the triangle to exist, we must have ...
1 ≥ sin(B)
1 ≥ b·sin(A)/a
a ≥ b·sin(A) . . . . . . . always true if a > b
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4)
a ? b·sin(A)
17 > 13·sin(63°) . . . . true (no calculator needed) triangle exists
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5)
a ? b·sin(A)
24 > 7·sin(120°) . . . . true (no calculator needed) triangle exists
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Additional comment
In general if b > a and the triangle exists, there will be two possible triangles with the given dimensions.
The only rational numbers that will give a right angle for B are ...
a=1, b=2, A=30° . . . . . and any multiples of these side lengths