在三角形ABC中，sinAsinBsin2C的最大值是什么？

nyy

在三角形ABC中，sinAsinBsin2C的最大值是什么？

nyy

sinAsin2Bsin3C的最大值呢？

nyy

@王守恩 来挑战这个问题！

王守恩

先找到答案。再去补为什么。

Table[NMaximize[{Sin[u*A] Sin[v*B] Sin[w*C], u*A + v*B + w*C == Pi, A > 0, B > 0, C > 0}, {A, B, C}], {u, 9}, {v, u, 9}, {w, v, 9}]

nyy

来呀@王守恩

王守恩

没找到规律。来个简单的先试试——在三角形ABC中, Sin[k*A]*Sin[k*B]*Sin[k*C]的最大值是什么？ 其中: A + B + C = Pi。 k = 1,2,3,4,5,6,...

Table[Maximize[{Sin[k*A]*Sin[k*B]*Sin[k*C], 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[3])/8, {A -> \[Pi]/3, B -> \[Pi]/3, C -> \[Pi]/3}},
k=2。{(3 Sqrt[3])/8, {A -> \[Pi]/3, B -> \[Pi]/3, C -> \[Pi]/3}},
k=3。{(3 Sqrt[3])/8, {A -> \[Pi]/9, B -> (4 \[Pi])/9, C -> (4 \[Pi])/9}},
k=4。{(3 Sqrt[3])/8, {A -> \[Pi]/6, B -> (5 \[Pi])/12, C -> (5 \[Pi])/12}},
k=5。{(3 Sqrt[3])/8, {A -> \[Pi]/15, B -> (7 \[Pi])/15, C -> (7 \[Pi])/15}},
k=6。{(3 Sqrt[3])/8, {A -> \[Pi]/9, B -> (4 \[Pi])/9, C -> (4 \[Pi])/9}},
k=7。{(3 Sqrt[3])/8, {A -> \[Pi]/21, B -> (10 \[Pi])/21, C -> (10 \[Pi])/21}},
k=8。{(3 Sqrt[3])/8, {A -> \[Pi]/12, B -> (11 \[Pi])/24, C -> (11 \[Pi])/24}},
k=9。{(3 Sqrt[3])/8, {A -> \[Pi]/27, B -> (13 \[Pi])/27, C -> (13 \[Pi])/27}}}
.....

上面的解也可以由下面的方程解得。
Table[Solve[{Sin[k*A]*Sin[a*B]^2 == (3 Sqrt[3])/8, A + 2 B == Pi, Pi > k*A > 0}, {A, B}], {k, 36}]

{{A -> \[Pi]/3, B -> \[Pi]/3}}, {{A -> \[Pi]/3, B -> \[Pi]/3}}, {{A -> \[Pi]/9, B -> (4 \[Pi])/9}},
{{A -> \[Pi]/6, B -> (5 \[Pi])/12}}, {{A -> \[Pi]/15, B -> (7 \[Pi])/15}}, {{A -> \[Pi]/9, B -> (4 \[Pi])/9}},
{{A -> \[Pi]/21, B -> (10 \[Pi])/21}}, {{A -> \[Pi]/12, B -> (11 \[Pi])/24}}, {{A -> \[Pi]/27, B -> (13 \[Pi])/27}},
{{A -> \[Pi]/15, B -> (7 \[Pi])/15}}, {{A -> \[Pi]/33, B -> (16 \[Pi])/33}}, {{A -> \[Pi]/18, B -> (17 \[Pi])/36}},
{{A -> \[Pi]/39, B -> (19 \[Pi])/39}}, {{A -> \[Pi]/21, B -> (10 \[Pi])/21}}, {{A -> \[Pi]/45, B -> (22 \[Pi])/45}},
{{A -> \[Pi]/24, B -> (23 \[Pi])/48}}, {{A -> \[Pi]/51, B -> (25 \[Pi])/51}}, {{A -> \[Pi]/27, B -> (13 \[Pi])/27}},
{{A -> \[Pi]/57, B -> (28 \[Pi])/57}}, {{A -> \[Pi]/30,  B -> (29 \[Pi])/60}}, {{A -> \[Pi]/63, B -> (31 \[Pi])/63}},
{{A -> \[Pi]/33, B -> (16 \[Pi])/33}}, {{A -> \[Pi]/69, B -> (34 \[Pi])/69}}, {{A -> \[Pi]/36, B -> (35 \[Pi])/72}},
{{A -> \[Pi]/75, B -> (37 \[Pi])/75}}, {{A -> \[Pi]/39, B -> (19 \[Pi])/39}}, {{A -> \[Pi]/81, B -> (40 \[Pi])/81}},
{{A -> \[Pi]/42, B -> (41 \[Pi])/84}}, {{A -> \[Pi]/87, B -> (43 \[Pi])/87}}, {{A -> \[Pi]/45, B -> (22 \[Pi])/45}},
{{A -> \[Pi]/93, B -> (46 \[Pi])/93}}, {{A -> \[Pi]/48, B -> (47 \[Pi])/96}}, {{A -> \[Pi]/99, B -> (49 \[Pi])/99}},
{{A -> \[Pi]/51, B -> (25 \[Pi])/51}}, {{A -> \[Pi]/105, B -> (52 \[Pi])/105}}, {{A -> \[Pi]/54, B -> (53 \[Pi])/108}}}

问:  A的通项公式是什么？  B的通项公式是什么？

nyy

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[k*A]*Sin[k*B]*Sin[k*C]的最大值是什么？ 其中: A +  ...

你这不正确

王守恩

不好玩！找个慢慢长大的——在三角形ABC中, Sin[k*A]*Sin[B]*Sin[B]的最大值是什么？ 其中: A + B + B = Pi。 k = 1,2,3,4,5,6,...

Table[NMaximize[{Sin[k*A] Sin[B] Sin[B], 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[Pi/(2 + 4 k^2)] Cos[(k*Pi)/(2 + 4 k^2)]^2, {k, 32}]

nyy

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[k*A]*Sin*Sin的最大值是什么？ 其中: A + B + B = Pi。 k = ...

不要弄数值解

nyy

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[k*A]*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！！！