Suppose a triangle has two sides of length 42 and 35, and that the angle between these two sides is 120. What is the length of the third side of the triangle?

Question

mathmale · Answer

@hanzi:  are you sure about using Heron's formula here?  Isn't that formula for finding AREA?  Want to think this through and perhaps propose a proper method of finding the length of the third side of the triangle?

anonymous · Answer

sorry, just slipped my mind, its law of sines

anonymous · Answer

ok so what the formula?

anonymous · Answer

formula is same, just change of name

anonymous · Answer

use law of sines formula, sina/A=sinb/B=sinc/C 
A=42, B=35, c=120
 put in above equation,
 also use a+b+c=180 (sum of angles of triangles) 
youll get value of C, the third side

mathmale · Answer

@Hanzi:  I'm afraid that 's not true.  If you believe the Law of Sines applies here, demonstrate it.  There is another "Law" that is more relevant to this problem.

anonymous · Answer

what would be the other Law that i could use?

mathmale · Answer

Please look up "Law of Cosines."  Once you've read about that and copied down the formula, I'd be happy to help you apply it to this particular homework problem.

anonymous · Answer

ok i will reply back when i have.

mathmale · Answer

I like your attitude.  Go for it!

anonymous · Answer

c2=a2+b2-2ab cos(C)? is that right?

anonymous · Answer

would the answer be 66.78?

mathmale · Answer

You have the correct formula.  Just checking:  What does C stand for in this case?  c?  What are the lengths of a and b?

mathmale · Answer

$$IF:  ~c^2=a^2+b^2-2ab*\cos C,~then~here:~c^2=35^2+42^2-2(35)(42)\cos(120)$$

mathmale · Answer

Please calculate this.