Question

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.

Answer 1

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=180-C = 180 - 32-28-30 = 147-31-30 = 1967π/2400 radians
a+b=21.46+46.28=67.74
a-b=21.46-46.28=-24.82

By the law of tangents,
tan(A-B)/tan(A+B)=(a-b)/(a+b)
or
tan(A-B)=tan(A+B)(a-b)/(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(-(a-b)) = 1.394801777162711
log(a+b) = 1.830845192308612
log(tan(A+B)*(a-b)/(a+b) = -0.1962309+1.3948018-1.8308452
=-0.6322743
Antilog(-0.6322743)=0.2331985 as before.
Convert 0.2331985 to degrees,
A-B=13-21-41, and
A+B=147-31-30
A = (160-53-11)/2=80-26-36
B = (134-09-49)/2=67-04-55
side c can be found by the cosine rule or the sine rule.
Check my work.

Answer 2

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:
(a-b)/(a+b) = tan((A-B)/2)/tan((A+B)/2)

Using the previous values,
a=21.46
b=46.28
C=32-28-30=32.475
We have
A+B=180-32.475=147.525
a+b=67.74
a-b=-24.82

Now apply the tangent rule:
tan((A+B)/2)=3.433633
tan((A-B)/2)=tan((A+B)/2)*(a-b)/(a+b)
= -1.2580863
(A-B)/2 = atan(1.2580863)= -51.520285
A-B = -103.04057

A=(147.525-103.04057)/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.