Solve a Triangle

[nycmathguy]

Section 6.2

Can you tell by looking at the following problems which of these may lead to two triangles, two solutions? If you can, please solve that particular problem in step by step fashion?

 

[MathLover1]

a=sqrt(b^2+c^2-2bc*cos(A))

b=sqrt(a^2+c^2-2ac*cos(B))

c=sqrt(a^2+b^2-2ab*cos(C))

33.
given: A=24° , a=4, b=18

a=sqrt(b^2+c^2-2bc*cos(A))
4=sqrt(324+c^2-36c *cos(24°) )
16=324+c^2-36c *0.9135454576426
16=c^2-32.8877c +324

c=sqrt(4^2+18^2-2*4*18*cos(C))
c=sqrt(340 - 144cos(C))
c^2=340 - 144cos(C)

substitute in

16=(340 - 144cos(C))^2-32.8877sqrt(340 - 144cos(C)) +324

no real solution

35. A=42°, B=35°, c=1.2

C=180-(42+35)=103°

a=sqrt(b^2 - 1.78355 b + 1.44)

a^2=b^2 - 1.78355 b + 1.44

substitute in

b=sqrt(a^2+c^2-2ac*cos(B))
b=sqrt(b^2 - 1.78355 b + 1.44+1.2^2-2(b^2 - 1.78355 b + 1.44)*1.2*cos(35))
b=sqrt(-0.965965 b^2 + 1.72285 b + 0.0490105)
b^2=-0.965965 b^2 + 1.72285 b + 0.0490105
b^2+0.965965 b^2 - 1.72285 b - 0.0490105=0

b≈-0.028-> disregard negative solution for side
b=0.9

then
a^2=b^2 - 1.78355 b + 1.44

a^2=(0.904)^2 - 1.78355*0.904 + 1.44
a^2=0.6448868
a=sqrt(0.6448868)
a=0.8

use same way to solve 36

 

[nycmathguy]

a=sqrt(b^2+c^2-2bc*cos(A))

b=sqrt(a^2+c^2-2ac*cos(B))

c=sqrt(a^2+b^2-2ab*cos(C))

33.
given: A=24° , a=4, b=18

a=sqrt(b^2+c^2-2bc*cos(A))
4=sqrt(324+c^2-36c *cos(24°) )
16=324+c^2-36c *0.9135454576426
16=c^2-32.8877c +324

c=sqrt(4^2+18^2-2*4*18*cos(C))
c=sqrt(340 - 144cos(C))
c^2=340 - 144cos(C)

substitute in

16=(340 - 144cos(C))^2-32.8877sqrt(340 - 144cos(C)) +324

no real solution

35. A=42°, B=35°, c=1.2

C=180-(42+35)=103°

a=sqrt(b^2 - 1.78355 b + 1.44)

a^2=b^2 - 1.78355 b + 1.44

substitute in

b=sqrt(a^2+c^2-2ac*cos(B))
b=sqrt(b^2 - 1.78355 b + 1.44+1.2^2-2(b^2 - 1.78355 b + 1.44)*1.2*cos(35))
b=sqrt(-0.965965 b^2 + 1.72285 b + 0.0490105)
b^2=-0.965965 b^2 + 1.72285 b + 0.0490105
b^2+0.965965 b^2 - 1.72285 b - 0.0490105=0

b≈-0.028-> disregard negative solution for side
b=0.9

then
a^2=b^2 - 1.78355 b + 1.44

a^2=(0.904)^2 - 1.78355*0.904 + 1.44
a^2=0.6448868
a=sqrt(0.6448868)
a=0.8

use same way to solve 36

Thank you. I will work out 36 on my next day off. Back to work tonight.

P. S. I will text a PM about a lease situation. Hoping you can help to clarify the matter.