Expert: Sherman D. - 11/28/2006

Question
Solve the following oblique triangles:

9.  a=31, b=15, c=17.
10. a=23.47, B=115°30',C=20°29'
11. a=134.2, b=84.54, B=52°9'11"
12. a=627.7, b=412.2, A=66°47'  

Answer
using these formula

Law of Cosines
a^2 = b^2 + c^2 - 2bc(cosA)
b^2 = a^2 + c^2 - 2ac(cosB)
c^2 = a^2 + b^2 - 2ab(cosC)

9.)

31^2 = 15^2 + 17^2 - 2(15 * 17)cos(A)
961 = 225 + 289 - 2(255)cosA
961 = 514 - 510cosA
447 = -510cosA
(-447/510) = cosA
cosA = (-149/170)
A = 151°13'10.2 or about 151.22°

15^2 = 31^2 + 17^2 - 2(31 * 17)cos(B)
225 = 961 + 289 - 2(527)cos(B)
225 = 1250 - 1054cosB
-1025 = -1054cosB
cosB = (1025/1054)
B = about 13.472° or 13°28'17.53

17^2 = 31^2 + 15^2 - 2(31 * 15)cos(C)
289 = 961 + 225 - 2(465)cos(C)
289 = 1186 - 930cosC
-897 = -930cosC
cosC = (897/930)
C = about 15.31° or 15°18'32.27

A = 151°13'10.2"
B = 13°28'17.53"
C = 15°18'32.27"

-----------------------------------------

10.)
Law of Sines
(a/sinA) = (b/sinB) = (c/sinC)

A + B + C = 180
A + 115°30' + 20°29' = 180
A + 135°59' = 180
A = 44°1'

(23.47/sin(44°1')) = b/(sin(115°30'))
b = (23.47sin(115°30'))/(sin(44°1'))
b = about 30.49

(23.47/sin(44°1')) = c/(sin(20°29'))
c = (23.47sin(20°29))/(sin(44°1'))
c = about 11.82

A = 44°1'
b = about 30.49
c = about 11.82

--------------------------

11.)

(84.54/sin(52°9'11") = (134.2/sin(A))
sin(A) = (134.2)/(84.54/sin(52°9'11"))
sin(A) = (134.2sin(52°9'11))/(84.54)

Unless i have done something wrong, i get A to be unknown

----------------------------

12.)

(627.7/sin(66°47')) = (412.2)/(sin(B))
sinB = (412.2)/(627.7/sin(66°47'))
sinB = (412.2sin(66°47'))/(627.7)
B = 37°7'16.92"

66°47' + 37°7'16.92" + C = 180
C = 76°5'43.08

(627.7/sin(66°47)) = c/(sin(76°5'43.08))
c = (627.7sin(76°5'43.08))/(sin(66°47))
c = 663.0

B = 37°7'16.92"
C = 76°5'43.08
c = about 663.0

Info found at http://www.clarku.edu/~djoyce/trig/oblique.html

It took me a while to finish these because i kept making little mistakes that through me off. As far as 9, 10, and 12, i have already checked using c^2 = a^2 + b^2 - 2ab(cos(C)), and they are correct, is just 11 i am not 100% on.

For further help, i recommend checking out answers.yahoo.com