The Cosine Law
For a triangle, with sides of length a, b and c, and angle q opposite the side of length c,
the cosine law say c2 = a2 + b2 - 2ab cos(q).
Proof of the Cosine Law
Introduce a coordinate system so that the origin z = 0 is at the angle and the ends of the adjacent sides have coordinates [x1,y1] and [x2,y2]. Then the following diagram and computation of the dot product implies
and computation of the dot product implies
x1x2+y1y2 = r1r2 cos( q) = ab cos( q)
But
c2 = (x1 - x2)2 + (y1 - y2)2 = (x12 - 2x1x2 + x22 ) + (y12 - 2y1y2 + y22 )
= (x12 + y12 ) + (x22 + y22 ) - 2(x1x2 + y1y2 )
= a2 + b2 - 2ab cos( q)
Pythagorean Theorem Converse:
If the sides of a triangle have length a, b and c with c2 = a2 + b2 then the triangle has a right angle opposite the side of length c.
Proof of Converse:
When 0 < q < 180,
cos(q) = 0 when and only when q = 90 degrees
Therefore c2 = a2 + b2 - 2ab cos(q) = a2 + b2 with a > 0, b >0 and 0 < q < 180 implies cos(q) = 0 and hence q = 90 degrees. So the triangle is a right triangle.
|
Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically