When Should You Use the Law of Sines and Cosines?

[havechanged]

Hey! I just wanted to post what I was getting confused on, concerning Sin and the whole Soh Cah Toa thing.

The Law of Siness is sinA = sinB= sinC
---- ---- ----
a b c

and the other one was actually the Law of Cosines, which is:

c squared= a squared + b squared - 2ab cosC

cos C= a squared+ b squared- c squared divided by 2ab.

What I still don't quite know is when these laws need to be used...

Thanks!

 

[HallsofIvy]

You'd do better to use parentheses:

The sine law is (sin A)/a= (sin B)/b= (sin C)/c
where A, B, C are the measures of angles in a triangle and a, b, c are the lengths of the sides opposite those angles.

The cosine law is c²= a²+ b²- 2ab cos(C) with similar laws for the other angles. Did you notice that if C= 90 degrees, cos(C)= 0 so this becomes the Pythagorean theorem?

Solving for cos(C), it would be better to write

cos(C)= (a²+ b²-c²)/(2ab) so there is no ambiguity.

As to when you use them, the answer is- when they work! In order to be able to "solve for" one variable in a formula, you have to know all the others. The cosine law works very nicely to solve for angles if you know the lengths of the sides (SSS congruence), or to solve for one side if you know the lengths of the other two sides and the angle between them (SAS). The sine law works to solve for one side if you know one side and the angles at either end (ASA) or to solve for an angle if you know two sides and the angle between them (SAS again but now solving for the angle).

 

[havechanged]

Thanks!