Question

mathematica计算方程组：

帮个忙，计算一下。这软件我没下载，根据这个方程组求出角度1、角度2的表达式，其中c（？）表示余弦，s（？）表示正弦

Answer 1

稍作简化后得：
θ1 = -1. ArcCos[(1/( a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 + 2.` x0^3 + 2.` x0 y0^2 - 0.5√(y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 - 32.` x0^2 y0^2 - 16.` y0^4 + a2^2 (32.` x0^2 + 32.` y0^2) + a1^2 (32.` a2^2 + 32.` x0^2 + 32.` y0^2))))],
θ2 = -1.` ArcCos[( 0.`- 0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/a2]},
以下将符号\[Theta]和\[Sqrt]替换成 θ 和 √ 后可得结果，\[VeryThinSpace]为很小空格（即表示相乘）。
{\[Theta]1 -> -1.` ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 +
2.` x0^3 + 2.` x0 y0^2 -
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 +
32.` y0^2))))], \[Theta]2 ->
ArcCos[(0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/
a2]}, {\[Theta]1 ->
ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 + 2.` x0^3 +
2.` x0 y0^2 -
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 +
32.` y0^2))))], \[Theta]2 -> -1.` ArcCos[(
0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/
a2]}, {\[Theta]1 ->
ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 + 2.` x0^3 +
2.` x0 y0^2 -
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 + 32.` y0^2))))], \[Theta]2 ->
ArcCos[(
0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/
a2]}, {\[Theta]1 -> -1.` ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 +
2.` x0^3 + 2.` x0 y0^2 +
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 +
32.` y0^2))))], \[Theta]2 -> -1.` ArcCos[(
0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/
a2]}, {\[Theta]1 -> -1.` ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 +
2.` x0^3 + 2.` x0 y0^2 +
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 +
32.` y0^2))))], \[Theta]2 ->
ArcCos[(0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/
a2]}, {\[Theta]1 ->
ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 + 2.` x0^3 +
2.` x0 y0^2 +
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 +
32.` y0^2))))], \[Theta]2 -> -1.` ArcCos[(
0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/
a2]}, {\[Theta]1 ->
ArcCos[(1/(
a1 (4.` x0^2 + 4.` y0^2)))(2.` a1^2 x0 - 2.` a2^2 x0 + 2.` x0^3 +
2.` x0 y0^2 +
0.5` \[Sqrt](y0^2 (-16.` a1^4 - 16.` a2^4 - 16.` x0^4 -
32.` x0^2 y0^2 - 16.` y0^4 +
a2^2 (32.` x0^2 + 32.` y0^2) +
a1^2 (32.` a2^2 + 32.` x0^2 + 32.` y0^2))))], \[Theta]2 ->
ArcCos[(
0.`\[VeryThinSpace]-
0.5` a1 + (-0.5` a2^2 + 0.5` x0^2 + 0.5` y0^2)/a1)/a2]}}

追问

谢谢了。两个都正确，只能给先答那个朋友分咯

Answer 2

用t表示θ,
方程为Solve[{a2 Cos[t1 + t2] + a1 Cos[t1] == x0, a2 Sin[t1 + t2] + a1 Sin[t1] == y0}, {t1, t2}]
解得:
{{t2 -> -ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(2 a1 a2)],
t1 -> -ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 - \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 -> -ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(
2 a1 a2)],
t1 -> ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 - \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 -> -ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(
2 a1 a2)],
t1 -> -ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 + \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 -> -ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(
2 a1 a2)],
t1 -> ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 + \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 ->
ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(2 a1 a2)],
t1 -> -ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 - \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 ->
ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(2 a1 a2)],
t1 -> ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 - \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 ->
ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(2 a1 a2)],
t1 -> -ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 + \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}, {t2 ->
ArcCos[(-a1^2 - a2^2 + x0^2 + y0^2)/(2 a1 a2)],
t1 -> ArcCos[(a1^3 x0 - a1 a2^2 x0 + a1 x0^3 +
a1 x0 y0^2 + \[Sqrt](-a1^6 y0^2 + 2 a1^4 a2^2 y0^2 -
a1^2 a2^4 y0^2 + 2 a1^4 x0^2 y0^2 + 2 a1^2 a2^2 x0^2 y0^2 -
a1^2 x0^4 y0^2 + 2 a1^4 y0^4 + 2 a1^2 a2^2 y0^4 -
2 a1^2 x0^2 y0^4 - a1^2 y0^6))/(2 (a1^2 x0^2 +
a1^2 y0^2))]}}