In triangle ABC with tan(B/2) + tan(C/2) = 1, prove that tan(A) ≥ 24/7.
clearvars,clc
for b=1:179
tb=tand(b/2);
c=2*atand(1-tb);
if c<0
c=c+180;
end
a=180-b-c;
if a>0
ta=tand(a);
fprintf('%3d %9.6f %11.7f %11.7f %11.7f\n',b,c,a,ta,ta*7);
end
end
disp(' ')
for b=52:.01:54
tb=tand(b/2);
c=2*atand(1-tb);
if c<0
c=c+180;
end
a=180-b-c;
if a>0
ta=tand(a);
fprintf('%5.2f %9.6f %11.7f %11.7f %11.7f\n',b,c,a,ta,ta*7);
end
end
To show tan(A)*7 doesn't dip below 24:
b° c a tan(a) tan(a)*7
1 89.497799 89.5022008 115.0952818 805.6669723
2 88.991119 89.0088808 57.8034021 404.6238147
3 88.479842 88.5201581 38.7088920 270.9622438
4 87.963846 88.0361542 29.1638679 204.1470753
5 87.443007 87.5569929 23.4387586 164.0713099
6 86.917198 87.0828015 19.6237115 137.3659806
7 86.386290 86.6137102 16.9002229 118.3015602
8 85.850147 86.1498525 14.8590447 104.0133129
9 85.308635 85.6913653 13.2728207 92.9097448
10 84.761611 85.2383888 12.0051407 84.0359848
11 84.208933 84.7910668 10.9692017 76.7844116
12 83.650453 84.3495468 10.1071383 70.7499679
13 83.086020 83.9139803 9.3788932 65.6522527
14 82.515478 83.4845223 8.7558574 61.2910020
15 81.938668 83.0613322 8.2170541 57.5193787
16 81.355426 82.6445735 7.7467539 54.2272772
17 80.765586 82.2344139 7.3329317 51.3305221
18 80.168974 81.8310256 6.9662381 48.7636667
19 79.565414 81.4345855 6.6392951 46.4750656
20 78.954725 81.0452751 6.3462037 44.4234262
21 78.336719 80.6632809 6.0821922 42.5753452
22 77.711206 80.2887944 5.8433596 40.9035175
23 77.077988 79.9220122 5.6264874 39.3854117
24 76.436863 79.5631365 5.4288968 38.0022779
25 75.787625 79.2123749 5.2483419 36.7383930
26 75.130059 78.8699407 5.0829258 35.5804806
27 74.463947 78.5360530 4.9310372 34.5172601
28 73.789063 78.2109373 4.7912989 33.5390925
29 73.105175 77.8948249 4.6625285 32.6376996
30 72.412046 77.5879538 4.5437056 31.8059395
31 71.709431 77.2905686 4.4339466 31.0376265
32 70.997079 77.0029206 4.3324834 30.3273840
33 70.274732 76.7252683 4.2386464 29.6705250
34 69.542123 76.4578772 4.1518506 29.0629540
35 68.798980 76.2010204 4.0715837 28.5010861
36 68.045022 75.9549785 3.9973971 27.9817797
37 67.279960 75.7200399 3.9288972 27.5022803
38 66.503499 75.4965012 3.8657393 27.0601748
39 65.715333 75.2846671 3.8076216 26.6533515
40 64.915149 75.0848508 3.7542812 26.2799687
41 64.102626 74.8973740 3.7054895 25.9384265
42 63.277433 74.7225675 3.6610493 25.6273451
43 62.439229 74.5607709 3.6207922 25.3455455
44 61.587667 74.4123333 3.5845764 25.0920347
45 60.722387 74.2776132 3.5522847 24.8659932
46 59.843022 74.1569785 3.5238237 24.6667660
47 58.949193 74.0508071 3.4991221 24.4938549
48 58.040513 73.9594869 3.4781306 24.3469143
49 57.116584 73.8834158 3.4608212 24.2257487
50 56.176998 73.8230020 3.4471875 24.1303124
51 55.221336 73.7786641 3.4372445 24.0607116
52 54.249169 73.7508312 3.4310298 24.0172089
53 53.260057 73.7399428 3.4286043 24.0002299
54 52.253551 73.7464493 3.4300533 24.0103733
55 51.229189 73.7708115 3.4354890 24.0484233
56 50.186499 73.8135008 3.4450523 24.1153664
57 49.125001 73.8749993 3.4589159 24.2124113
58 48.044201 73.9557994 3.4772879 24.3410151
59 46.943596 74.0564041 3.5004163 24.5029141
60 45.822674 74.1773262 3.5285946 24.7001625
61 44.680912 74.3190884 3.5621685 24.9351794
62 43.517777 74.4822232 3.6015439 25.2108075
63 42.332728 74.6672719 3.6471978 25.5303845
64 41.125215 74.8747847 3.6996903 25.8978322
65 39.894680 75.1053197 3.7596810 26.3177672
66 38.640558 75.3594423 3.8279487 26.7956406
67 37.362275 75.6377247 3.9054162 27.3379137
68 36.059255 75.9407448 3.9931832 27.9522826
69 34.730915 76.2690850 4.0925668 28.6479676
70 33.376669 76.6233312 4.2051556 29.4360893
71 31.995928 77.0040717 4.3328807 30.3301647
72 30.588105 77.4118952 4.4781097 31.3467679
73 29.152610 77.8473896 4.6437751 32.5064257
74 27.688860 78.3111399 4.8335498 33.8348484
75 26.196274 78.8037262 5.0520934 35.3646539
76 24.674278 79.3257215 5.3054043 37.1378304
77 23.122311 79.8776894 5.6013338 39.2093368
78 21.539819 80.4601811 5.9503576 41.6525035
79 19.926268 81.0737324 6.3667684 44.5673787
80 18.281139 81.7188610 6.8705834 48.0940840
81 16.603937 82.3960626 7.4907247 52.4350730
82 14.894192 83.1058076 8.2705836 57.8940852
83 13.151463 83.8485367 9.2783555 64.9484884
84 11.375343 84.6246572 10.6277093 74.3939650
85 9.565462 85.4345382 12.5232601 87.6628210
86 7.721494 86.2785061 15.3742516 107.6197609
87 5.843160 87.1568397 20.1356016 140.9492112
88 3.930235 88.0697650 29.6720881 207.7046167
89 1.982550 89.0174502 58.3076452 408.1535162
90 0.000000 90.0000000 Inf Inf
52.00 54.249169 73.7508312 3.4310298 24.0172089
52.01 54.239362 73.7506377 3.4309867 24.0169070
52.02 54.229554 73.7504459 3.4309440 24.0166077
52.03 54.219744 73.7502558 3.4309016 24.0163111
52.04 54.209933 73.7500673 3.4308596 24.0160171
52.05 54.200119 73.7498806 3.4308180 24.0157258
52.06 54.190304 73.7496956 3.4307767 24.0154371
52.07 54.180488 73.7495123 3.4307359 24.0151511
52.08 54.170669 73.7493306 3.4306954 24.0148677
52.09 54.160849 73.7491507 3.4306553 24.0145870
52.10 54.151028 73.7489725 3.4306156 24.0143090
52.11 54.141204 73.7487960 3.4305762 24.0140336
52.12 54.131379 73.7486211 3.4305373 24.0137608
52.13 54.121552 73.7484480 3.4304987 24.0134907
52.14 54.111723 73.7482766 3.4304605 24.0132233
52.15 54.101893 73.7481068 3.4304227 24.0129586
52.16 54.092061 73.7479388 3.4303852 24.0126964
52.17 54.082228 73.7477725 3.4303481 24.0124370
52.18 54.072392 73.7476078 3.4303115 24.0121802
52.19 54.062555 73.7474449 3.4302752 24.0119261
52.20 54.052716 73.7472837 3.4302392 24.0116746
52.21 54.042876 73.7471242 3.4302037 24.0114258
52.22 54.033034 73.7469664 3.4301685 24.0111797
52.23 54.023190 73.7468102 3.4301337 24.0109362
52.24 54.013344 73.7466558 3.4300993 24.0106954
52.25 54.003497 73.7465031 3.4300653 24.0104572
52.26 53.993648 73.7463521 3.4300317 24.0102217
52.27 53.983797 73.7462028 3.4299984 24.0099889
52.28 53.973945 73.7460553 3.4299655 24.0097587
52.29 53.964091 73.7459094 3.4299330 24.0095313
52.30 53.954235 73.7457652 3.4299009 24.0093064
52.31 53.944377 73.7456227 3.4298692 24.0090843
52.32 53.934518 73.7454820 3.4298378 24.0088648
52.33 53.924657 73.7453430 3.4298069 24.0086480
52.34 53.914794 73.7452056 3.4297763 24.0084338
52.35 53.904930 73.7450700 3.4297460 24.0082223
52.36 53.895064 73.7449361 3.4297162 24.0080135
52.37 53.885196 73.7448039 3.4296868 24.0078073
52.38 53.875327 73.7446734 3.4296577 24.0076039
52.39 53.865455 73.7445446 3.4296290 24.0074031
52.40 53.855582 73.7444175 3.4296007 24.0072049
52.41 53.845708 73.7442922 3.4295728 24.0070095
52.42 53.835831 73.7441685 3.4295452 24.0068167
52.43 53.825953 73.7440466 3.4295181 24.0066266
52.44 53.816074 73.7439264 3.4294913 24.0064391
52.45 53.806192 73.7438078 3.4294649 24.0062544
52.46 53.796309 73.7436911 3.4294389 24.0060723
52.47 53.786424 73.7435760 3.4294133 24.0058929
52.48 53.776537 73.7434626 3.4293880 24.0057162
52.49 53.766649 73.7433510 3.4293632 24.0055421
52.50 53.756759 73.7432410 3.4293387 24.0053707
52.51 53.746867 73.7431328 3.4293146 24.0052020
52.52 53.736974 73.7430263 3.4292909 24.0050360
52.53 53.727078 73.7429216 3.4292675 24.0048727
52.54 53.717182 73.7428185 3.4292446 24.0047120
52.55 53.707283 73.7427172 3.4292220 24.0045540
52.56 53.697382 73.7426175 3.4291998 24.0043987
52.57 53.687480 73.7425196 3.4291780 24.0042461
52.58 53.677577 73.7424234 3.4291566 24.0040962
52.59 53.667671 73.7423290 3.4291356 24.0039489
52.60 53.657764 73.7422362 3.4291149 24.0038044
52.61 53.647855 73.7421452 3.4290946 24.0036625
52.62 53.637944 73.7420559 3.4290748 24.0035233
52.63 53.628032 73.7419683 3.4290553 24.0033868
52.64 53.618118 73.7418825 3.4290361 24.0032530
52.65 53.608202 73.7417984 3.4290174 24.0031218
52.66 53.598284 73.7417160 3.4289991 24.0029934
52.67 53.588365 73.7416353 3.4289811 24.0028676
52.68 53.578444 73.7415563 3.4289635 24.0027445
52.69 53.568521 73.7414791 3.4289463 24.0026242
52.70 53.558596 73.7414036 3.4289295 24.0025065
52.71 53.548670 73.7413298 3.4289131 24.0023915
52.72 53.538742 73.7412577 3.4288970 24.0022792
52.73 53.528813 73.7411874 3.4288814 24.0021695
52.74 53.518881 73.7411188 3.4288661 24.0020626
52.75 53.508948 73.7410519 3.4288512 24.0019584
52.76 53.499013 73.7409868 3.4288367 24.0018569
52.77 53.489077 73.7409234 3.4288226 24.0017580
52.78 53.479138 73.7408617 3.4288088 24.0016619
52.79 53.469198 73.7408017 3.4287955 24.0015684
52.80 53.459257 73.7407435 3.4287825 24.0014777
52.81 53.449313 73.7406870 3.4287699 24.0013896
52.82 53.439368 73.7406322 3.4287578 24.0013043
52.83 53.429421 73.7405792 3.4287459 24.0012216
52.84 53.419472 73.7405279 3.4287345 24.0011416
52.85 53.409522 73.7404783 3.4287235 24.0010644
52.86 53.399570 73.7404304 3.4287128 24.0009898
52.87 53.389616 73.7403843 3.4287026 24.0009180
52.88 53.379660 73.7403400 3.4286927 24.0008488
52.89 53.369703 73.7402973 3.4286832 24.0007824
52.90 53.359744 73.7402564 3.4286741 24.0007186
52.91 53.349783 73.7402172 3.4286654 24.0006576
52.92 53.339820 73.7401798 3.4286570 24.0005992
52.93 53.329856 73.7401441 3.4286491 24.0005436
52.94 53.319890 73.7401101 3.4286415 24.0004906
52.95 53.309922 73.7400779 3.4286343 24.0004404
52.96 53.299953 73.7400474 3.4286276 24.0003929
52.97 53.289981 73.7400187 3.4286212 24.0003481
52.98 53.280008 73.7399917 3.4286151 24.0003060
52.99 53.270034 73.7399664 3.4286095 24.0002666
53.00 53.260057 73.7399428 3.4286043 24.0002299
53.01 53.250079 73.7399210 3.4285994 24.0001960
53.02 53.240099 73.7399010 3.4285950 24.0001647
53.03 53.230117 73.7398827 3.4285909 24.0001362
53.04 53.220134 73.7398661 3.4285872 24.0001103
53.05 53.210149 73.7398512 3.4285839 24.0000872
53.06 53.200162 73.7398382 3.4285810 24.0000668
53.07 53.190173 73.7398268 3.4285784 24.0000491
53.08 53.180183 73.7398172 3.4285763 24.0000341
53.09 53.170191 73.7398093 3.4285746 24.0000219
53.10 53.160197 73.7398032 3.4285732 24.0000123
53.11 53.150201 73.7397988 3.4285722 24.0000055
53.12 53.140204 73.7397962 3.4285716 24.0000014
53.13 53.130205 73.7397953 3.4285714 24.0000000 min
53.14 53.120204 73.7397961 3.4285716 24.0000013
53.15 53.110201 73.7397987 3.4285722 24.0000054
53.16 53.100197 73.7398031 3.4285732 24.0000122
53.17 53.090191 73.7398092 3.4285745 24.0000217
53.18 53.080183 73.7398170 3.4285763 24.0000339
53.19 53.070173 73.7398266 3.4285784 24.0000488
53.20 53.060162 73.7398380 3.4285809 24.0000665
53.21 53.050149 73.7398510 3.4285838 24.0000869
53.22 53.040134 73.7398659 3.4285871 24.0001100
53.23 53.030118 73.7398825 3.4285908 24.0001358
53.24 53.020099 73.7399008 3.4285949 24.0001644
53.25 53.010079 73.7399209 3.4285994 24.0001957
53.26 53.000057 73.7399427 3.4286042 24.0002297
53.27 52.990034 73.7399663 3.4286095 24.0002665
53.28 52.980008 73.7399916 3.4286151 24.0003060
53.29 52.969981 73.7400187 3.4286212 24.0003482
53.30 52.959952 73.7400476 3.4286276 24.0003931
53.31 52.949922 73.7400782 3.4286344 24.0004408
53.32 52.939889 73.7401105 3.4286416 24.0004912
53.33 52.929855 73.7401446 3.4286492 24.0005444
53.34 52.919820 73.7401805 3.4286572 24.0006002
53.35 52.909782 73.7402181 3.4286656 24.0006589
53.36 52.899743 73.7402574 3.4286743 24.0007202
53.37 52.889701 73.7402986 3.4286835 24.0007843
53.38 52.879659 73.7403415 3.4286930 24.0008511
53.39 52.869614 73.7403861 3.4287030 24.0009207
53.40 52.859568 73.7404325 3.4287133 24.0009930
53.41 52.849519 73.7404806 3.4287240 24.0010680
53.42 52.839469 73.7405305 3.4287351 24.0011458
53.43 52.829418 73.7405822 3.4287466 24.0012263
53.44 52.819364 73.7406356 3.4287585 24.0013096
53.45 52.809309 73.7406908 3.4287708 24.0013956
53.46 52.799252 73.7407478 3.4287835 24.0014844
53.47 52.789194 73.7408065 3.4287966 24.0015759
53.48 52.779133 73.7408669 3.4288100 24.0016701
53.49 52.769071 73.7409292 3.4288239 24.0017671
53.50 52.759007 73.7409932 3.4288381 24.0018668
53.51 52.748941 73.7410589 3.4288528 24.0019693
53.52 52.738874 73.7411264 3.4288678 24.0020745
53.53 52.728804 73.7411957 3.4288832 24.0021825
53.54 52.718733 73.7412668 3.4288990 24.0022932
53.55 52.708660 73.7413396 3.4289152 24.0024067
53.56 52.698586 73.7414141 3.4289319 24.0025230
53.57 52.688510 73.7414905 3.4289488 24.0026419
53.58 52.678431 73.7415686 3.4289662 24.0027637
53.59 52.668352 73.7416485 3.4289840 24.0028882
53.60 52.658270 73.7417301 3.4290022 24.0030154
53.61 52.648187 73.7418135 3.4290208 24.0031454
53.62 52.638101 73.7418987 3.4290397 24.0032782
53.63 52.628014 73.7419856 3.4290591 24.0034137
53.64 52.617926 73.7420743 3.4290788 24.0035519
53.65 52.607835 73.7421648 3.4290990 24.0036930
53.66 52.597743 73.7422570 3.4291195 24.0038368
53.67 52.587649 73.7423510 3.4291405 24.0039833
53.68 52.577553 73.7424468 3.4291618 24.0041326
53.69 52.567456 73.7425444 3.4291835 24.0042847
53.70 52.557356 73.7426437 3.4292056 24.0044395
53.71 52.547255 73.7427448 3.4292282 24.0045971
53.72 52.537152 73.7428477 3.4292511 24.0047575
53.73 52.527048 73.7429523 3.4292744 24.0049206
53.74 52.516941 73.7430587 3.4292981 24.0050865
53.75 52.506833 73.7431669 3.4293222 24.0052552
53.76 52.496723 73.7432769 3.4293467 24.0054266
53.77 52.486611 73.7433886 3.4293715 24.0056008
53.78 52.476498 73.7435021 3.4293968 24.0057777
53.79 52.466383 73.7436174 3.4294225 24.0059575
53.80 52.456266 73.7437345 3.4294486 24.0061400
53.81 52.446147 73.7438533 3.4294750 24.0063253
53.82 52.436026 73.7439739 3.4295019 24.0065133
53.83 52.425904 73.7440963 3.4295292 24.0067041
53.84 52.415780 73.7442205 3.4295568 24.0068977
53.85 52.405654 73.7443464 3.4295849 24.0070941
53.86 52.395526 73.7444742 3.4296133 24.0072933
53.87 52.385396 73.7446037 3.4296422 24.0074952
53.88 52.375265 73.7447349 3.4296714 24.0076999
53.89 52.365132 73.7448680 3.4297011 24.0079074
53.90 52.354997 73.7450029 3.4297311 24.0081176
53.91 52.344861 73.7451395 3.4297615 24.0083307
53.92 52.334722 73.7452779 3.4297924 24.0085465
53.93 52.324582 73.7454181 3.4298236 24.0087651
53.94 52.314440 73.7455600 3.4298552 24.0089865
53.95 52.304296 73.7457038 3.4298872 24.0092107
53.96 52.294151 73.7458493 3.4299197 24.0094376
53.97 52.284003 73.7459967 3.4299525 24.0096674
53.98 52.273854 73.7461458 3.4299857 24.0098999
53.99 52.263703 73.7462966 3.4300193 24.0101352
54.00 52.253551 73.7464493 3.4300533 24.0103733
>>
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Posted by Charlie
on 2023-09-06 08:53:09 |
heuristic "proof"