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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
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-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
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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.
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www.whyslopes.com
Complex Numbers
Hint: See the (newest) Complex Number. Starter Lesson. for a simple geometric introduction, then continue with easy consequence below.
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
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