Expert: Sherman D. - 10/6/2008

Question
Hello. I need to solve the following case III triangle using the half-angle formula, here is the triangle

a = 27, b = 21 and c = 24

I am using a skeleton solution so it would be of great help if your answers were in that form. Here is what I've done so far.

Skeleton Solution

a = 27
b = 21
c = 24
2s = 72
s = 36
s-a = 9
s-b = 15
s-c = 12
log (s-a) = 0.954242509
log (s-b) = 1.176091259
log (s-c) = 1.079181246
           3.209515014
log s = 1.556302501
      2) 1.653212513
log r = 0.826606256


log r = 0.826606256
log (s-a) = 0.954242509
log tan A/2 = ?
A = ?(in form of degrees, min, and sec)
log r = 0.82660
log (s-b) = 1.17609
log tan B/2 = ?
B = ?(degrees, min, and sec)
log r = 0.82660
log (s-c) = 1.07918
log tan C/2 = ?
C = ?(degrees, min, and sec)

I know that tan A/2 = r/s-a, but it doesn't add up for me. Please correct me on any errors you find, I'm sure there are many. Thanks for the help!

Answer
a = 27, b = 21 and c = 24

2s = 27 + 21 + 24
2s = 72
s = 36

log(36 - 27) = log(9) = 0.954242509
log(36 - 21) = log(15) = 1.176091259
log(36 - 24) = log(12) = 1.079181246

log(9) + log(15) + log(12) = 3.209515015, not 3.209515014 when your rounding.

log(36) = 1.556302501

log(r) = 0.826606257, not 0.826606256

log(tan(A/2)) = log(sqrt(5)/3)
A = 73°23'54.42"
log(tan(B/2)) = log(sqrt(5)/5)
B = 48°11'22.87"
log(tan(C/2)) = log(sqrt(5)/4)
C = 58°24'42.71"

if you add A, B, and C up, you will get 180.

The mistake i made was forgetting to square root the ((s - a)(s - b)(s - c))/s) to get r. But then i corrected it.

Just one question, couldn't you had used

tan(A/2) = sqrt((s - b)(s - c)/(s(s - a)))
tan(B/2) = sqrt((s - a)(s - c)/(s(s - b)))
tan(C/2) = sqrt((s - a)(s - b)/(s(s - c)))

In otherwords, did you have to do it using logarithms.

if you have any questions, just let me know.

by the way, to get the minutes and seconds, just subtract the whole number, then multiply by 60, then subtract the whole number of that, and multiply by 60.

also

log(tan(A/2)) = log(r/(s - a)) = log(r) - log(s - a)

if that helps.

if you have adobe media reader, you can find more info at http://lessons.educationdirect.com/pdf/6232d.pdf