This is an outline that i have to use to find the solution of the question. It involves logarithms and half-angle formulas.I have worked though it but got stuck on some parts. I'd like if you look over my work.
solve the triangle for which given parts are
a=27 ,b=21 ,c=24
Now i used the cosine law and got the answers
A=73 deg 23' 55"
B=48 deg 11' 23"
C=58 deg 24' 43"
...................
okay now for this outline;
a=27
b=21
c=25
----
2s=73
s=36.5
s-a=9.5
s-b=15
s-c=11.5
log(s-a)= .97772
log(s-b)=1.19033
log(s-c)=1.06070
--------
=3.22875
log s =1.56229
--------
2)1.66646
--------
log r =0.83323
now this part I'm having trouble with.
log r = 0.83323
log(s-a) = 0.97772
--------
log tan A/2 = 9.85551 - 10
A = ? << can you help me convert.
log r = 0.83323
log (s-b) = 1.19033
--------
log tan B/2 = 9.64290 - 10
B = ? same here my brain freezes
log r = 0.83323
log (s-c) = 1.06070
--------
log tan C/2 = 9.77253 - 10
C = ?
A+B+C= 180 deg 0' 2"
Let's go through each part step by step.
1. First, you correctly used the cosine law to find the angles A, B, and C of the triangle.
2. Now, let's move on to the second part of your outline. You have given the values of sides a, b, and c as 27, 21, and 25 respectively. You have also correctly calculated parameter s, where s = (a + b + c)/2 = (27 + 21 + 25)/2 = 73/2 = 36.5.
3. Next, you calculated the differences s-a, s-b, and s-c, which are 9.5, 15, and 11.5 respectively. These differences will be used to calculate the logarithms in the next steps.
4. To find log(s-a), log(s-b), and log(s-c), you took the logarithm of each difference. Based on the values you provided, log(s-a) = 0.97772, log(s-b) = 1.19033, and log(s-c) = 1.06070.
5. Next, you added the logarithms log(s-a), log(s-b), and log(s-c) together. log(s-a) + log(s-b) + log(s-c) = 0.97772 + 1.19033 + 1.06070 = 3.22875.
6. You then calculated log(s), where s is equal to the semiperimeter of the triangle. Based on your values, log(s) = 1.56229.
7. Now, you need to calculate log(r) by subtracting log(s) from the previous sum. log(r) = log(s-a) + log(s-b) + log(s-c) - log(s) = 3.22875 - 1.56229 = 1.66646.
8. The value you calculated for log(r) is approximately 1.66646.
9. Now, let's move on to the part you are having trouble with, which involves finding the values of A, B, and C using the half-angle formulas and logarithms.
a) For angle A, you need to find log(tan(A/2)). You have log(r) = 0.83323 and log(s-a) = 0.97772. To find log(tan(A/2)), you subtract log(s-a) from log(r): log(tan(A/2)) = log(r) - log(s-a) = 0.83323 - 0.97772 = -0.14449.
To convert log(tan(A/2)) to the angle A, you need to take the antilogarithm (10 to the power of the value). So, A = tan^(-1)(10^(-0.14449)). Using a calculator, you can find the value of A.
b) Similarly, for angle B, you find log(tan(B/2)) by subtracting log(s-b) from log(r): log(tan(B/2)) = log(r) - log(s-b) = 0.83323 - 1.19033 = -0.35710. To find the value of B, you use B = tan^(-1)(10^(-0.35710)).
c) For angle C, you find log(tan(C/2)) by subtracting log(s-c) from log(r): log(tan(C/2)) = log(r) - log(s-c) = 0.83323 - 1.06070 = -0.22747. To find the value of C, you use C = tan^(-1)(10^(-0.22747)).
10. Finally, you can confirm your calculations by adding A, B, and C. A + B + C should equal 180 degrees.
I hope this explanation helps you understand the steps involved and how to proceed with the remaining calculations.