help me abut this question plz,,,,,,,,?

Question

the side (a) of a triangle ABC is calculated from the following formula
a/(sinA) = b/(sinB)
where lower-case letters refer to the sides and upper-case letters refer to the angles opposite those sides
The side b in error by 3.1%.
Aand B are given as 52 degrees and 51 degrees respectively, to the nearest degree.
Using partial differentiation find the greatest percentage error in the calculated value of a.

Answer 1

a/(sinA) = b/(sinB)
a = b(sinA)/(sinB)
da = ((sinA)/(sinB))db + (b(cosA)/(sinB))dA/A - (b(sinA)(cosB)/(sin^2B))dB/B

da/a = (sinB)/(b(sinA)) ((sinA)/(sinB))db + (sinB)/(b(sinA)) (b(cosA)/(sinB))dA/A - (sinB)/(b(sinA)) (b(sinA)(cosB)/(sin^2B))dB/B

da/a = db/b + ((cosA)/(sinA))dA/A - ((cosB)/(sinB))dB/B
da/a = db/b + dA/(AtanA) - dB/(BtanB)
da/a = (± 0.031) + (± 0.5/52)/tan52 - (± 0.5/51)/tan51
da/a ≈ ± 0.031 ± 0.00751 ± 0.00794
max da/a ≈ 0.046451 ≈ 4.65%

check:
Let b = 100
a = 100sin52/sin51
a = 101.39803752301331877380449471538
a(max) = 103.1sin52.5/sin50.5
a(max) = 106.0032 ===> 4.54%
a(min) = 96.9sin51.5/sin51.5
a(min) = 96.9 ==> 4.44%

I no longer remember the rationale for using dA/A and dB/B instead of dA and dB, except that it is necessary for dimensional consistency and it produces a reasonable answer.