三角函数方程

northwolves

$cosx+ 4 sinx+ 3 cosxsinx=0$ 有没有什么巧妙的算法？如何得出解析解？


$x1=2n\pi+2.5544$

$x2=2n\pi-0.1425$

zeroieme

半角替换成四次方程

1. Cos[x]+4 Sin[x]+3 Cos[x] Sin[x]==0/.x->2ArcTan[t]//TrigExpand//Solve[#,t,Reals]&(*限制实数解*)//ToRadicals//x->2ArcTan[t]+2n \[Pi]/.#&

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结果
\(\left\{x\to 2 \pi  n+2 \tan ^{-1}\left(\frac{1}{2} \left(-\sqrt{2 \left(3 \sqrt{5}-1\right)}+\sqrt{5}+1\right)\right),x\to 2 \pi  n+2 \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3 \sqrt{5}-1\right)}+\frac{\sqrt{5}}{2}+\frac{1}{2}\right)\right\}\)

wayne

1. {x,y}/.Solve[{x+4 y+3x y==0,x^2+y^2==1},{x,y}]//FullSimplify//ToRadicals

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\[\left\{\frac{1}{3} \left(\sqrt{5}-\sqrt{2 \sqrt{5}+3}-2\right),\frac{1}{6} \left(\sqrt{5}+\sqrt{2 \left(5 \sqrt{5}-9\right)}-1\right)\right\},\]
\[\left\{\frac{1}{3} \left(\sqrt{5}+\sqrt{2 \sqrt{5}+3}-2\right),\frac{1}{6} \left(\sqrt{5}-3 \sqrt{\frac{10 \sqrt{5}}{9}-2}-1\right)\right\},\]
\[\left\{\frac{1}{3} \left(-\sqrt{5}-i \sqrt{2 \sqrt{5}-3}-2\right),\frac{1}{6} \left(-\sqrt{5}+i \sqrt{2 \left(5 \sqrt{5}+9\right)}-1\right)\right\},\]
\[\left\{\frac{1}{3} \left(-\sqrt{5}+i \sqrt{2 \sqrt{5}-3}-2\right),\frac{1}{6} \left(-\sqrt{5}-i \sqrt{2 \left(5 \sqrt{5}+9\right)}-1\right)\right\}\]

northwolves

软件里好解决。手工算有没有什么技巧？

zeroieme

northwolves 发表于 2020-10-2 20:17
软件里好解决。手工算有没有什么技巧？

计算量不大。

northwolves

$-79 - 88 x - 10 x^2 + 24 x^3 + 9 x^4=0$