The a side of a triangle ABC is calculated from the following formula?

Question

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 1.1%.
Aand B are given as 25 degrees and 58 degrees respectively, to the nearest degree.
Using partial differentiation find the greatest percentage error in the calculated value of a.

Answer 1

a = bsinA/sinB
∆a/a = ∆b/b = 1.1%
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Partial differentiation means only one variable can be changed. Therefore, A and B are constants.

Answer 2

Solving for a, you get
a = b sin A/sin B

So the differential formula is
da = (sin A/sin B) db + (b cos A/sin B) dA - (b sin A cos B / (sin B)^2) dB
This is where the partial derivatives come in. The coefficients of db, dA, dB are the partial derivatives of a with respect to b, A, B.

Dividing the differential formula by the original formula for a, you get

da/a = db/b + (cos A / sin A) dA - (cos B / sin B) dB
= db/b + (cos 25/sin 25) dA - (cos 58/sin 58) dB
= db/b + (2.1445) dA - (0.6249) dB

You are given that the max error of b is 0.011, that is db/b = 0.011.
And the max error of A and B is plus or minus 1 degree, which must be converted to radians.
1 degree = pi/180 radians = 0.01745
To maximize da/a, choose dA = 0.01745 and dB = -0.01745.

da/a = 0.011 + (2.1445)(0.01745) - (0.6249)(-0.01745)
= 0.011 + 0.03742 + 0.0109
= 0.05932
= 5.9 %

EDIT: Since the angles are measured to the nearest degree, this probably means that the maximum error is half a degree, rather than 1 degree as I used above. I'll leave the recalculation up to you.

Answer 3

As far as I understand there are three variables to be considered
a= b/sinB *sinA
1) You must express angles in radians
da = sinA/sinB *db +b/sinB *cosA*dA +b*sinA(-1/sin^2B*cosB)*dB
db/b=1.1/100 so I think you need the value of b
dA and dB are 2pi/360 rad
You get the bound of the percentage error dividing by the calculated value of a provided you got b and taking all summands positive