在三角形ABC中，sinAsinBsin2C的最大值是什么？
在三角形ABC中，sinAsinBsin2C的最大值是什么？ sinAsin2Bsin3C的最大值呢？ @王守恩 来挑战这个问题！ 先找到答案。再去补为什么。Table Sin Sin, u*A + v*B + w*C == Pi, A > 0, B > 0, C > 0}, {A, B, C}], {u, 9}, {v, u, 9}, {w, v, 9}] 来呀@王守恩 没找到规律。来个简单的先试试——在三角形ABC中, Sin*Sin*Sin的最大值是什么？ 其中: A + B + C = Pi。 k = 1,2,3,4,5,6,...
Table*Sin*Sin, A + B + C == Pi, Pi/2 > A > 0, Pi/2 > B > 0, Pi/2 > C > 0}, {A, B, C}], {k, 9}]
k=1。{(3 Sqrt)/8, {A -> \/3, B -> \/3, C -> \/3}},
k=2。{(3 Sqrt)/8, {A -> \/3, B -> \/3, C -> \/3}},
k=3。{(3 Sqrt)/8, {A -> \/9, B -> (4 \)/9, C -> (4 \)/9}},
k=4。{(3 Sqrt)/8, {A -> \/6, B -> (5 \)/12, C -> (5 \)/12}},
k=5。{(3 Sqrt)/8, {A -> \/15, B -> (7 \)/15, C -> (7 \)/15}},
k=6。{(3 Sqrt)/8, {A -> \/9, B -> (4 \)/9, C -> (4 \)/9}},
k=7。{(3 Sqrt)/8, {A -> \/21, B -> (10 \)/21, C -> (10 \)/21}},
k=8。{(3 Sqrt)/8, {A -> \/12, B -> (11 \)/24, C -> (11 \)/24}},
k=9。{(3 Sqrt)/8, {A -> \/27, B -> (13 \)/27, C -> (13 \)/27}}}
.....
上面的解也可以由下面的方程解得。
Table*Sin^2 == (3 Sqrt)/8, A + 2 B == Pi, Pi > k*A > 0}, {A, B}], {k, 36}]
{{A -> \/3, B -> \/3}}, {{A -> \/3, B -> \/3}}, {{A -> \/9, B -> (4 \)/9}},
{{A -> \/6, B -> (5 \)/12}}, {{A -> \/15, B -> (7 \)/15}}, {{A -> \/9, B -> (4 \)/9}},
{{A -> \/21, B -> (10 \)/21}}, {{A -> \/12, B -> (11 \)/24}}, {{A -> \/27, B -> (13 \)/27}},
{{A -> \/15, B -> (7 \)/15}}, {{A -> \/33, B -> (16 \)/33}}, {{A -> \/18, B -> (17 \)/36}},
{{A -> \/39, B -> (19 \)/39}}, {{A -> \/21, B -> (10 \)/21}}, {{A -> \/45, B -> (22 \)/45}},
{{A -> \/24, B -> (23 \)/48}}, {{A -> \/51, B -> (25 \)/51}}, {{A -> \/27, B -> (13 \)/27}},
{{A -> \/57, B -> (28 \)/57}}, {{A -> \/30,B -> (29 \)/60}}, {{A -> \/63, B -> (31 \)/63}},
{{A -> \/33, B -> (16 \)/33}}, {{A -> \/69, B -> (34 \)/69}}, {{A -> \/36, B -> (35 \)/72}},
{{A -> \/75, B -> (37 \)/75}}, {{A -> \/39, B -> (19 \)/39}}, {{A -> \/81, B -> (40 \)/81}},
{{A -> \/42, B -> (41 \)/84}}, {{A -> \/87, B -> (43 \)/87}}, {{A -> \/45, B -> (22 \)/45}},
{{A -> \/93, B -> (46 \)/93}}, {{A -> \/48, B -> (47 \)/96}}, {{A -> \/99, B -> (49 \)/99}},
{{A -> \/51, B -> (25 \)/51}}, {{A -> \/105, B -> (52 \)/105}}, {{A -> \/54, B -> (53 \)/108}}}
问:A的通项公式是什么？B的通项公式是什么？ 王守恩 发表于 2025-11-13 16:37
没找到规律。来个简单的先试试——在三角形ABC中, Sin*Sin*Sin的最大值是什么？ 其中: A +...
你这不正确 不好玩！找个慢慢长大的——在三角形ABC中, Sin*Sin*Sin的最大值是什么？ 其中: A + B + B = Pi。 k = 1,2,3,4,5,6,...
Table Sin Sin, A + B + B == Pi, Pi > A > 0, Pi > B > 0}, {A, B}], {k, 9}]
{0.649519, {A -> 1.047198, B -> 1.04720}},
{0.869619, {A -> 0.695721, B -> 1.22294}},
{0.936528, {A -> 0.495557, B -> 1.32302}},
{0.963092, {A -> 0.380667, B -> 1.38046}},
{0.976008, {A -> 0.307953, B -> 1.41682}},
{0.983196, {A -> 0.258194, B -> 1.44170}},
{0.987591, {A -> 0.222124, B -> 1.45973}},
{0.990467, {A -> 0.194823, B -> 1.47338}},
{0.992450, {A -> 0.173460, B -> 1.48407}},
......
特别地把第一个数提出来——{0.649519, 0.869619, 0.936528, 0.963092, 0.976008, 0.983196, 0.987591, 0.990467, 0.992450, 0.993875, 0.994932, 0.995737, 0.996365, 0.996864, 0.997267, 0.997597,
0.997871, 0.9981, 0.998295, 0.998461, 0.998603, 0.998727, 0.998835, 0.99893, 0.999014, 0.999088, 0.999155, 0.999214, 0.999267, 0.999315, 0.999359, 0.999398,————慢慢向“1”靠拢。}
Table Cos[(k*Pi)/(2 + 4 k^2)]^2, {k, 32}] 王守恩 发表于 2025-11-14 11:07
不好玩！找个慢慢长大的——在三角形ABC中, Sin*Sin*Sin的最大值是什么？ 其中: A + B + B = Pi。 k = ...
不要弄数值解 王守恩 发表于 2025-11-14 11:07
不好玩！找个慢慢长大的——在三角形ABC中, Sin*Sin*Sin的最大值是什么？ 其中: A + B + B = Pi。 k = ...
至少2的时候有精确解！！！
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