Expert: Sherman D. - 3/6/2007

Question
Find the indicated angle theta. It has an obtuse triangle with 18 as the opposite 7 as the adjacent and 21 as the longest side and it gives me no angles?

Answer
Law of Cosine

a^2 = b^2 + c^2 - 2bc(cos(A))
b^2 = a^2 + c^2 - 2ac(cos(B))
c^2 = a^2 + b^2 - 2ab(cos(C))

or to make it simplier for you

A = cos^-1((b^2 + c^2 - a^2)/(2bc))
B = cos^-1((a^2 + c^2 - b^2)/(2ac))
C = cos^-1((a^2 + b^2 - c^2)/(2ab))

so

a = 18
b = 7
c = 21

A = cos^-1((b^2 + c^2 - a^2)/(2bc))
A = cos^-1((7^2 + 21^2 - 18^2)/(2(7)(21))
A = cos^-1((49 + 441 - 324)/294)
A = cos^-1(166/294)
A = 55°37'25.27"

B = cos^-1((a^2 + c^2 - b^2)/(2ac))
B = cos^-1((18^2 + 21^2 - 7^2)/(2 * 18 * 21)
B = cos^-1((324 + 441 - 49)/756)
B = cos^-1(716/756)
B = 18°43'17.37

C = cos^-1((a^2 + b^2 - c^2)/(2ab))
C = cos^-1((18^2 + 7^2 - 21^2)/(2(18)(7))
C = cos^-1((324 + 49 - 441)/252)
C = cos^-1(-68/252)
C = 105°39'17.3"

info found at www.math.com/tables/trig/identities.htm