In the scalene triangle ABC, from AC it is 2.3 km. From CB is 3.7 km. AB is the longest side but is not given. The only angle given is at point C which is 125°. How much longer the path is from ACB than straight from A to B? Which trigonometric law did you use?

Question

ranga · Answer

You are given two sides of a triangle and the included angle and we need to find the length of the side opposite the given angle.  The Law of Cosines will be the most useful one here:
c^2 = a^2 + b^2 - 2abcos(C) or 
AB^2 = BC^2 + AC^2 - 2(BC)(AC)cos(C)
Plug in the numbers:
AB^2 = (3.7)^2 + (2.3)^2 - 2(3.7)(2.3)cos(125) = ?
Take the square root to find AB.

ranga · Answer

How much longer the path is from ACB than straight from A to B?
Compute AC + BC - AB
2.3 + 3.7 - AB   (substitute AB found earlier)

anonymous · Answer

See I did all that but then you get left with cos(125 which is a negative number and that square rooted I ended with $$\sqrt{-1.124}$$ unless I should leave it at $$\sqrt{1.96\cos(125}$$

ranga · Answer

AB^2 = (3.7)^2 + (2.3)^2 - 2(3.7)(2.3)cos(125) 
The negative value of cos(125) will make - 2(3.7)(2.3)cos(125)  positive.

ranga · Answer

AB^2 = (3.7)^2 + (2.3)^2 - 2(3.7)(2.3)cos(125) 
(3.7)^2 = 13.69
(2.3)^2 =   5.29
2(3.7)(2.3)cos(125) = -9.76
AB^2 = 13.69 + 5.29 + 9.76 = 28.74
AB = sqrt(28.74) = 5.36

anonymous · Answer

Ah. That makes more sense. And should help me solve other problems with the law of cosine. Thank you!

ranga · Answer

You are welcome.  
Don't forget the second part to this question.