Without the calculator let's just estimate that .5/42 = 1.2% change in the angle; we reduce the length a=37 by 1.2% to get 37×.988=36.556. That's probably good enough to choose A.
Let's do it the way they want.
[tex]\dfrac{\sin C}{c} = \dfrac{\sin A}{a}[/tex]
[tex]c = \dfrac{a \sin C}{\sin A} = \dfrac{37 \sin 41.5}{\sin 42} = 36.6400[/tex]
Answer: A
Estimate worked. Sine is approximately linear when the range is small enough.
