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A cute little angle to find (Posted on 2006-01-18) Difficulty: 4 of 5
In triangle ABC with:
BC = 1;
angle B=45 degrees;
D is on AB such that DC =1;
E and H are on segment AC and EH = 1;
F is on segment AD such that EF=.5;
G is on segment CD;
EFGH is a rectangle.

Find angle A

See The Solution Submitted by Jer    
Rating: 3.6667 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Solution | Comment 5 of 7 |
(In reply to Solution by Bractals)

The derivation of
1 = [cos(A) - sin(A)]*[sqrt(2) - 2*sin(A)]
from equationa (1) and (2) could probably use some explanation. The answer does agree with my solution of
sin(A) = sin(45)*(1 - 1/(2*sin(45-A)))

The computer solution starts with a wild estimate of 30 degrees; it then iteratively evaluates the right side of the equation and takes the arcsin, to get a new estimate for A. After 135 iterations it has settled into an alternation between 7.362314630758033 and 7.362314630758032 degrees:

 1 -41.217452862549180   50   7.362340270639818   99   7.362314630727474
 2  20.657540712474300   51   7.362295220879728  100   7.362314630781166
 3  -8.663883646612330   52   7.362329324386308  101   7.362314630740522
 4  15.557932917728360   53   7.362303507405358  102   7.362314630771290
 5  -0.697041813913077   54   7.362323051338801  103   7.362314630747997
 6  12.302923572638010   55   7.362308256221321  104   7.362314630765630
 7   3.016154048009711   56   7.362319456399194  105   7.362314630752283
 8  10.286128470060280   57   7.362310977658034  106   7.362314630762386
 9   4.949042637521939   58   7.362317396221922  107   7.362314630754737
10   9.070955161152380   59   7.362312537250623  108   7.362314630760529
11   6.002873870368511   60   7.362316215581719  109   7.362314630756144
12   8.352919305453733   61   7.362313431017174  110   7.362314630759463
13   6.590665453578152   62   7.362315538983992  111   7.362314630756950
14   7.933870824668424   63   7.362313943214188  112   7.362314630758854
15   6.922478575097712   64   7.362315151241413  113   7.362314630757411
16   7.691151053803616   65   7.362314236742495  114   7.362314630758504
17   7.111025405315577   66   7.362314929035071  115   7.362314630757677
18   7.551191320075026   67   7.362314404956804  116   7.362314630758304
19   7.218557747625563   68   7.362314801693734  117   7.362314630757828
20   7.470696911693151   69   7.362314501356549  118   7.362314630758187
21   7.280013006926912   70   7.362314728717346  119   7.362314630757916
22   7.424472610040901   71   7.362314556601025  120   7.362314630758123
23   7.315176373188716   72   7.362314686896258  121   7.362314630757965
24   7.397951266819597   73   7.362314588260360  122   7.362314630758086
25   7.335309605361266   74   7.362314662929555  123   7.362314630757992
26   7.382742206050452   75   7.362314606403597  124   7.362314630758065
27   7.346841568400882   76   7.362314649194790  125   7.362314630758008
28   7.374022859881136   77   7.362314616801066  126   7.362314630758051
29   7.353448325668271   78   7.362314641323715  127   7.362314630758019
30   7.369024891315625   79   7.362314622759616  128   7.362314630758043
31   7.357233867627468   80   7.362314636812982  129   7.362314630758025
32   7.366160305259074   81   7.362314626174324  130   7.362314630758038
33   7.359403064917967   82   7.362314634227985  131   7.362314630758028
34   7.364518556941454   83   7.362314628131215  132   7.362314630758036
35   7.360646113093143   84   7.362314632746581  133   7.362314630758030
36   7.363577669370648   85   7.362314629252666  134   7.362314630758035
37   7.361358452768537   86   7.362314631897623  135   7.362314630758032
38   7.363038455426965   87   7.362314629895343  136   7.362314630758033
39   7.361766670806257   88   7.362314631411106  137   7.362314630758032
40   7.362729440083120   89   7.362314630263644  138   7.362314630758033
41   7.362000608723296   90   7.362314631132295  139   7.362314630758032
42   7.362552349110620   91   7.362314630474709  140   7.362314630758033
43   7.362134672354540   92   7.362314630972514  141   7.362314630758032
44   7.362450861894646   93   7.362314630595668  142   7.362314630758033
45   7.362211500870700   94   7.362314630880947  143   7.362314630758032
46   7.362392701758163   95   7.362314630664985  144   7.362314630758033
47   7.362255529437309   96   7.362314630828473  145   7.362314630758032
48   7.362359371474123   97   7.362314630704708  146   7.362314630758033
49   7.362280761166326   98   7.362314630798401  147   7.362314630758032

The program is:

DECLARE FUNCTION asin# (x#)
DEFDBL A-Z
pi = ATN(1) * 4
dr = pi / 180
a = 30
CLS
FOR i = 1 TO 49 * 3
 sa = SIN(45 * dr) * (1 - 1 / (2 * SIN((45 - a) * dr)))
 a = asin(sa) / dr
 r = (i - 1) MOD 49 + 1
 c = ((i - 1) \ 49) * 25 + 1
 LOCATE r, c
 PRINT USING "### ###.###############"; i; a;
NEXT

FUNCTION asin (x)
 asin = ATN(x / SQR(1 - x * x))
END FUNCTION

 


  Posted by Charlie on 2006-01-19 09:11:10
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