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When to Use Law of Sines or Law of Cosines

Date: 02/01/2006 at 14:07:32
From: Anisha
Subject: Sine Law and Cosine Law

When do we use the law of sines as opposed to the law of cosines? 

I thought that the sine law is used when we need to find an angle or a
side and two sides are given but the angle in between them is not given.

Date: 02/01/2006 at 20:04:44
From: Doctor Schwa
Subject: Re: Sine Law and Cosine Law

Yeah, that's the right idea, Anisha.

The law of cosines relates 3 sides and an angle.  So if you know 3
sides, you can use it, or if you know two sides and an angle, you can
find the third side.  However, because of the form of the equation, if
you have an angle that's not between the two sides, you get a
quadratic equation in that case--a bit messy.

So the law of cosines is best for SSS and SAS, and it's OK for SSA
as long as you're willing to solve a quadratic.

The law of sines relates two sides and the angles opposite them.  So
any time you have two angles (and then can easily figure out the
third), it's easy to use the law of sines: ASA or AAS.

In the SSA case, you can use the law of sines, but you have to
remember that when you get something like sin(alpha) = 0.7 or
whatever, there might be two possible triangles.  You have to check
alpha = 44.427 degrees or so, but it's also possible that alpha =
135.573 degrees or so.

So in the SSA case, the law of sines is easier, but you have to
remember to check for that second possibility; if you use the law of
cosines, you're stuck with the quadratic formula, but the +/- in the
quadratic formula immediately reminds you that there are two
possibilities that need to be checked.

In the other cases it's clear which one to use: SSS or SAS means the
law of cosines, ASA or AAS means the law of sines.

I hope that helps clear things up!

- Doctor Schwa, The Math Forum 
Associated Topics:
High School Trigonometry

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