The Geometry of the Law of Cosines

In the triangle ABC the side lengths are a = BC, b = CA, c = AB.  The point D is the foot of the altitude through B.  Let us call angle ACB = t. For each of these figures, write CD, BD, and AD in terms of trig functions of t and a, b, c.

CD =

CD =

BD =

BD =

AD =

AD =

Algebra of the Law of Cosines

Suppose that A = (a1, a2, a3), B = (b1, b2, b3), C = (0, 0, 0).  Use the algebraic distance formula to compute:


Dot Product and the Cosine

Write here what we found about the dot product and cosine

Pythagorean Theorem on the Sphere

Suppose that triangle ABC is a spherical triangle with a right angle at C.  We can choose space coordinates so that C = K = (0,0,1), A is in the (x,0,z) plane and B is in the (0,y,z) plane.  Here are two cross-sections, with I = (1, 0, 0) and J = (0, 1, 0).

 

The spherical distances a = BC, b = CA and c = AB are actually angle measures.  What are the angles in each case?

Using the angles a, b, c, what are the coordinates of A?

Using the angles a, b, c, what are the coordinates of B?

Now use the dot product to compute cos c.  This is the "Pythagorean Theorem for the Sphere."