根据你的两个式子,我用matlab求解结果如下,有兴趣的同学可以自行验证:
syms skewX skewY a b
res=solve(tan(skewX)==(sin(a)*sin(b)*cos(skewY)-cos(a)*sin(skewY))/(sin(a)*sin(b)*sin(skewY)+cos(a)*cos(skewY)),sin(a)*sin(b)*sin(skewY)+cos(a)*cos(skewY)==cos(b)*cos(skewY),a,b)
若令skewY=0,则结果是:
a的解为
-2*atan((((tan(skewX)^2 + 4)^(1/2) - 2)/tan(skewX))^(1/2))
2*atan((((tan(skewX)^2 + 4)^(1/2) + 2)/tan(skewX))^(1/2))
-2*atan((((tan(skewX)^2 + 4)^(1/2) + 2)/tan(skewX))^(1/2))
2*atan((-((tan(skewX)^2 + 4)^(1/2) - 2)/tan(skewX))^(1/2))
-2*atan((-((tan(skewX)^2 + 4)^(1/2) - 2)/tan(skewX))^(1/2))
b的解为
-2*atan((((tan(skewX)^2 + 4)^(1/2) - 2)/tan(skewX))^(1/2))
-2*atan((((tan(skewX)^2 + 4)^(1/2) + 2)/tan(skewX))^(1/2))
2*atan((((tan(skewX)^2 + 4)^(1/2) + 2)/tan(skewX))^(1/2))
-2*atan((-((tan(skewX)^2 + 4)^(1/2) - 2)/tan(skewX))^(1/2))
2*atan((-((tan(skewX)^2 + 4)^(1/2) - 2)/tan(skewX))^(1/2))
(第n行的a和第n行的b为一个解,即该方程共有5个解,看起来与之前二楼的计算过程差不多)
若没有令skewY为0,则结果非常长:
a的解为
-2*atan(((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) - 1))/(tan(skewY/2) + 1))
2*atan(((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) - 1))/(tan(skewY/2) + 1))
2*atan(((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) + 1))/(tan(skewY/2) - 1))
-2*atan(((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) + 1))/(tan(skewY/2) - 1))
2*atan(((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) + 1))/(tan(skewY/2) - 1))
-2*atan(((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) + 1))/(tan(skewY/2) - 1))
-2*atan(((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) - 1))/(tan(skewY/2) + 1))
2*atan(((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2)*(tan(skewY/2) - 1))/(tan(skewY/2) + 1))
b的解为
-2*atan((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
2*atan((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
-2*atan((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
2*atan((((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
-2*atan((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
2*atan((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) - 2*tan(skewY/2)*tan(skewX) + 2*tan(skewY/2)^4 + 2*tan(skewY/2)^3*tan(skewX) + 2)/(2*tan(skewY/2) + tan(skewX) + 2*tan(skewY/2)*tan(skewX) + 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 - 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
-2*atan((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))
2*atan((-((tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + tan(skewY/2)^2*(tan(skewX)^2 - 2*tan(skewY/2)^2*tan(skewX)^2 + tan(skewY/2)^4*tan(skewX)^2 - 4*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 4*tan(skewY/2)^4 + 4*tan(skewY/2)^3*tan(skewX) + 4)^(1/2) + 2*tan(skewY/2)*tan(skewX) - 2*tan(skewY/2)^4 - 2*tan(skewY/2)^3*tan(skewX) - 2)/(2*tan(skewY/2) + tan(skewX) - 2*tan(skewY/2)*tan(skewX) - 4*tan(skewY/2)^2 + 2*tan(skewY/2)^3 + 2*tan(skewY/2)^3*tan(skewX) - tan(skewY/2)^4*tan(skewX)))^(1/2))