"The laws of mathematics are not merely human inventions or creations. They simply 'are'; they exist quite independently of the human intellect. The most that any man with a keen intellect can do is to find out that they are there and to take cognizance of them." - (M.C. Escher)
Archimedes (287-212 B.C.) and the Sand Reckoner: The concept of infinity perplexed mathematicians until modern times. People in ancient days experienced a feeling of the infinite by gazing at heavenly bodies, and a grains of sand on a beach. They incorrectly linked the concept of infinity to extremely large numbers. In his writing Sand Reckoner, Archimedes dealt with extremely large numbers, such as (108)108 and suggested the concept of infinity. Interestingly, in this writing, he tossed aside the idea the the number of grains of sand on a beach are infinite, and he even determined a method for calculating the number on all the beaches of the earth.
How did Herkimer think cowboys from the old west sent letters that contained nothing but lies?
Answer: By phony express.
Herky wants to know:
If you get a new car for your spouse, should that be considered a good trade?
If you record your age in dog years, will you die sooner?
Does an agnostic pagan doubt the existence of many gods?
Reading: Review Sections in Chapter 13, as necessary.
Written: Pages 804-806/56, 63.
Page 812/54, 55.
The assigned problems represent realistic applications of the Law of Sines and the Law of Cosines. These important laws are summarized below:
Over the years humans have constructed various mnemonic devices to remember the number p to many decimal places. In 1914, the Scientific American published the mnemonic statement below. Replacing each word by the number of letters it contains yields p correct to 12 decimal places.
See, I have a rhyme assisting my feeble
brain, its tasks ofttimes resisting.
The Law of Sines (page 799): If the vertices of a triangle are A, B, and C, and if a, b, and c are the lengths of the sides opposite the respective vertices, then the Law of Sines says
sin A/a = sin B/b = sin C/c
The Law of Cosines (page 807): If the vertices of a triangle are A, B, and C, and if a, b, and c are the lengths of the sides opposite the respective vertices, then the Law of Cosines says
c2 = a2 + b2 - 2ab(cos C)
Also, the area of a triangle can be found by taking 1/2 of the product of the lengths of two sides times the sine of the included angle (page 802). The area of the triangle described above can be calculated by
(1/2)(ac)(sin B) = (1/2)(ab)(sin C) = (1/2)(bc)(sin A)
Problem: Two sides of a triangle are 10 inches and 12 inches. The angle included between the sides 114o. Find the area of the triangle.
Solution (with communication): The requested area is
(1/2)(10)(12)sin 114o = 54.81 (sq. in.)
Problem: The area of a triangle is 65 sq. cm. If two of the sides are 20 cm. and 12 cm., what is the measure of the angle between these two sides?
Solution (with communication): If x is the requested measure, then
(1/2)(20)(12)sin(x) = 65
==> sin (x) = 65/120.
One possible measure is x = sin-1(65/120) = 32.8o.
It is also possible that the triangle is obtuse and that the angle is 180o - 32.8o = 147.2o.