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Dear Ed,
The math puzzle of the week shows three triangles seemingly combined to form a 22-27-30 triangle. We are asked what is wrong with the figure. It is correct that the sum of the three angles around the interior point is greater than one revolution. However, I believe that the reason given is inadequate.
A simpler statement of the problem would be that we are given three second quadrant angles A, B, and C with cos A = -5/32, cos B = -179/224 and cos C = -151/322 and we are asked to show that A + B + C is not exactly one revolution. In the explanation given the three cosines have their rational numbers replaced by decimal approximations, then three inverse cosines are found as decimal approximations, these approximations are added and compared with a decimal approximation of 2 pi. I believe that after seven decimal approximations an exact or precise match cannot be required. Do you agree or disagree?
The puzzle is analogous to a problem solved in every high school trigonometry class. We are given a triangle with three sides of integral length. We use law of cosines to find rational numbers representing the cosines of all three angles independently of each other. We replace these three rational numbers with decimal approximations, then find three inverse cosines as decimal approximations. As a check, we compare the sum of these three angles to a decimal approximation of pi. In some of these problems the match is not exact. However, in no case do we ask what is wrong with the figure. It is a triangle given three sides. If the length of the largest side is less than the sum of the lengths of the smaller two sides, there can be nothing wrong with the figure.
I believe that the math puzzle of the week must be explained by a method
which is exact and precise throughout. I propose the following:
Use (sin x)^2 = 1 - (cos x)^2 to find the sines of the three angles
exactly.
If cos A = -5/32, then sin A = (3*sqrt(111))/32.
If cos B = -179/224, then sin B = (3*sqrt(2015))/224.
If cos C = -151/322, then sin C = (3*sqrt(8987))/322.
Use formulas for cos(A + B) and sin (A + B) to create a formula for
cos (A + B + C).
cos (A + B + C) = cos([ A + B ] + C).
cos (A + B + C) = + cos A * cos B * cos C
- sin A * sin B * cos C
- sin A * cos B * sin C
- cos A * sin B * sin C.
Find the exact value of the four right member terms.
+ cos A * cos B * cos C = - (135145)/2308096.
- sin A * sin B * cos C = + (1359*sqrt(223665))/2308096.
- sin A * cos B * sin C = + (1611*sqrt(997557))/2308096.
- cos A * sin B * sin C = + (45*sqrt(18108805))/2308096.
Add these four right members to find cos (A + B + C).
cos (A + B + C) does not equal 1.
A + B + C is not precisely one revolution.
Note that there is no mention of decimal approximation, radian measure
or degree measure in this method.
Is cos (A + B + C) greater than or less than 1?
Not required!
Is A + B + C greater than or less than one revolution?
Not required!
Do you believe that this exact and precise mathod is not required? Please reply.
Sincerely,
Robert H. Becker
P. S.
If you are curious, note the following:
cos (A + B + C) = (2308095.999999998603016)/2308096.
cos (A + B + C) = 0.999999999999999