Posted by Anon on .
Use logarithms and the law of tangents to solve the triangle ABC, given that a=21.46 ft, b=46.28 ft, and C=32°28'30"
Also Give checks.
I know the use of logarithms is unnecessary but i have to show my solution by using logarithms and that's the only reason I'm asking for help. Logarithms confuse me. Please help me.

trigonometry 
MathMate,
If you are not already familiar with the law of tangents, here's an article that can help you:
http://en.wikipedia.org/wiki/Law_of_tangents
We will be using the same notations in the following solution.
In the given case, sides a,b are known, and the included angle C. So the sum of angles
A+B=180C = 180  322830 = 1473130 = 1967π/2400 radians
a+b=21.46+46.28=67.74
ab=21.4646.28=24.82
By the law of tangents,
tan(AB)/tan(A+B)=(ab)/(a+b)
or
tan(AB)=tan(A+B)(ab)/(a+b)
=0.2331984
In log (to base 10), cannot calculate the values of negative numbers, so we will keep track of the sign ourselves:
log(tan(A+B)) = 0.1962309
log((ab)) = 1.394801777162711
log(a+b) = 1.830845192308612
log(tan(A+B)*(ab)/(a+b) = 0.1962309+1.39480181.8308452
=0.6322743
Antilog(0.6322743)=0.2331985 as before.
Convert 0.2331985 to degrees,
AB=132141, and
A+B=1473130
A = (1605311)/2=802636
B = (1340949)/2=670455
side c can be found by the cosine rule or the sine rule.
Check my work. 
trigonometry  correction 
MathMate,
Anon & Drws, thank you for pointing out there is a problem with this solution.
The tangent rule formula that I used was not correct. The formula used for the tangent rule is simply:
(ab)/(a+b) = tan((AB)/2)/tan((A+B)/2)
Using the previous values,
a=21.46
b=46.28
C=322830=32.475
We have
A+B=18032.475=147.525
a+b=67.74
ab=24.82
Now apply the tangent rule:
tan((A+B)/2)=3.433633
tan((AB)/2)=tan((A+B)/2)*(ab)/(a+b)
= 1.2580863
(AB)/2 = atan(1.2580863)= 51.520285
AB = 103.04057
A=(147.525103.04057)/2=22.242215°
B=(147.525+103.04057)/2=125.282785°
c can be found by the sine rule or the cosine rule, since 5 of the six unknowns have been calculated. I get c=30.44083 using the cosine rule and it checks with the sine rule.
I believe you can now handle the logarithm part. If you need further help, just post.
I apologize again for the unforgivable mistake in applying the tangent rule formula.