Laws of Cosines and Sines

Consider a triangle with sides a, b, and c.  The angles are A, B, and C such that A is opposite a, B is opposite b, and C is opposite c.  If you know any three facts -- lengths of sides or measures of angles -- then you can use one of these laws to find the lengths of all sides and measures of all angles.  These cases boil down to three cases:

SAS -- Side, included Angle, and Side (Law of Cosines)
SSA -- Two sides, and a non-included Angle (Law of Sines)
AAS -- Any two angles, and one side (Law of Sines)

Here is a breakdown of these three cases:

SAS -- Law of Cosines

The Law of Cosines gives the length of the side opposite an angle if you know the lengths of the other two sides.  This is the SAS case -- you know the Side, Angle, and Side.

a² = b² + c² - 2bc cos A

If you know the lengths of all three sides, you can solve for any angle:

A = acos((b²+c²-a²)/(2bc))
B = acos((c²+a²-b²)/(2ca))
C = acos((a²+b²-c²)/(2ab))

SSA -- Law of Sines

The Law of Sines is a/(sin A) = b/(sin B) = c/(sin C) = the diameter of the circumscribed circle.  (proof)

If you know the length of two sides and an angle other than the angle between those sides, then the Law of Sines can be used.  This is the "SSA" case -- Side, Side, Angle.  Assuming you know the lengths of sides a and b, and angle A,

(sin B)/b = (sin A)/a
sin B = (b sin A)/a
B = asin((b sin A)/a)

Usually, there are two different angles that can satisfy this equation, one in which B is acute, and the other in which B is obtuse, and these two angles are supplementary.  Both possible angles should be checked to make sure the sum of angles A, B, and C don't exceed pi.

AAS -- Law of Sines

If you know any two angles and any side, then you really know all three angles, so you have the "AAS" case -- Angle, Angle, Side.

(sin B)/b = (sin A)/a
b/sin B = a/sin A
b = (a sin B)/sin A

Related pages in this website:

Law of Sines Proof

The webmaster and author of the Math Help site is Graeme McRae.
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