By Joseph V. Collins

Excerpt from An ordinary Exposition of Grassmann's Ausdehnungslehre, or idea of Extension

The sum qf any variety of vectors is located through becoming a member of the start element of the second one vector to the top aspect of the 1st, the start aspect of the 3rd to the tip aspect of the second one. etc; the vector from the start element of the 1st vector to the top aspect of the final is the sum required.

The sum and distinction of 2 vectors are the diagonals of the parallelogram whose adjoining aspects are the given vectors.

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**Sample text**

We’re told that cos (x) = 0. So the Sun at an angle of elevation of x degrees has zero overness. The Sun is either directly overhead or directly below. The Basics of “Circle-ometry” 27 By adding an extra z degrees to the angle of elevation, we get an overness of 12 . There are two possible locations for the Sun with this overness. 1/2 The angle of smallest possible measure that shifts the Sun from one of the overhead/under-head positions to one of these new positions is the angle I’ve marked. All I need is the measure of that angle.

To get a sense of why sin (x) ≈ x if x is close to zero (and given in radians), consider a polygon with n sides inscribed in a circle of radius 1. O M A B The Graphs of Sine and Cosine in Radians 41 Let O be the center of the circle, A and B two adjacent vertices of the polygon, and M the midpoint of the side AB. We have that m∠AO B = 2π n (it’s one nth of a full turn) and so m∠M O B = πn . Looking at M O B we see B M = sin πn , and so AB = 2 sin πn , and the perimeter of the polygon is 2n sin πn .

180 = Radian Measure 33 Comment. Here is the MAA AMC question of the previous section in its original form: (#15, AMC 12A, 2006): Suppose cos (x) = 0 and cos (x + z) = 12 . What is the smallest possible positive value of z? (A) π 6 (B) π 3 (C) π 2 (D) 5π 6 (E) Care to rethink its solution in terms of radians? 4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 7 –360 –270 –180 45 90 –90 –1 34 180 270 360 The Graphs of Sine and Cosine in Degrees 35 Comment.