Муодилаи тригонометриро ҳал кунед:
\(\cos^{-1}3t-6\cos 3t=4\sin 3t\).
Ҳал.
\(\frac{1}{\cos 3t}-6\cos 3t=4\sin 3t\quad |\cdot\cos 3t\)
\(\cos 3t\cdot(\frac{1}{\cos 3t}-6\cos 3t)=4\sin 3t\cdot\cos 3t\)
\(\cos 3t\cdot\frac{1}{\cos 3t}-\cos 3t\cdot6\cos 3t=4\sin 3t\cos 3t\)
\(1-6\cos^{2}3t=4\sin 3t\cos 3t\)
\(4\sin 3t\cos 3t=1-6\cos^{2}3t\)
\(4\sin 3t\cos 3t+6\cos^{2}3t-1=0\)
\(4\sin 3t\cos 3t+6\cos^{2}3t-\)
\(\quad-(\sin^{2}3t+\cos^{2}3t)=0\)
\(4\sin 3t\cos 3t+6\cos^{2}3t-\)
\(\quad-\sin^{2}3t-\cos^{2}3t=0\)
\(4\sin 3t\cos 3t+5\cos^{2}3t-\sin^{2}3t=0 |\cdot\frac{1}{\cos^{2}3t}\)
\(\frac{1}{\cos^{2}3t}\cdot(4\sin 3t\cos 3t+5\cos^{2}3t-\)
\(\quad\quad\quad-\sin^{2}3t)=0\)
\(\frac{1}{\cos^{2}3t}\cdot4\sin 3t\cos 3t+\frac{1}{\cos^{2}3t}\cdot5\cos^{2}3t-\)
\(\quad-\frac{1}{\cos^{2}3t}\cdot\sin^{2}3t=0\)
\(\frac{4\sin 3t\cos 3t}{\cos^{2}3t}+\frac{5\cos^{2}3t}{\cos^{2}3t}- \frac{\sin^{2}3t}{\cos^{2}3t}=0\)
\(4\cdot\frac{\sin 3t}{\cos 3t}+5- (\frac{\sin 3t}{\cos 3t})^{2}=0\)
\(4\operatorname{tg}3t+5-\operatorname{tg}^{2}3t=0\quad |\cdot(-1)\)
\(\operatorname{tg}^{2}3t-4\operatorname{tg}3t-5=0\).
Бигзор \(x = \operatorname{tg}3t\). Пас,
\(x^2-4x-5=0\)
\(D=b^2-4\cdot a\cdot c=(-4)^2-4\cdot 1\cdot(-5)=\)
\(\quad=16+20=36=6^2\), \(D>0\)
\(x_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-(-4)\pm \sqrt{6^2}}{2\cdot1}=\frac{4\pm 6}{2}\)
\(x_{1}=\frac{4-6}{2}=\frac{-2}{2}=-1\);
\(x_{2}=\frac{4+6}{2}=\frac{10}{2}=5\).
Якум ҳолатро дида мебароем:
\(\operatorname{tg}3t=-1\)
\(3t=\operatorname{arctg}(-1)+\pi n\quad |\cdot\frac{1}{3}\)
\(\frac{1}{3}\cdot3t=\frac{1}{3}\cdot(\operatorname{arctg}(-1)+\pi n)\)
\(t=\frac{1}{3}\cdot\operatorname{arctg}(-1)+\frac{1}{3}\cdot\pi n\)
\(t=\frac{1}{3}\cdot(-\frac{\pi}{4})+\frac{\pi n}{3}\)
\(t=-\frac{\pi}{12}+\frac{\pi n}{3}\)
\(t=\frac{\pi n}{3}-\frac{\pi}{12}\)
\(t=\frac{\pi}{12}(4n-1)\).
Ҳолати дуюмро дида мебароем:
\(\operatorname{tg}3t=5\)
\(3t=\operatorname{arctg}5+\pi n\quad |\cdot\frac{1}{3}\)
\(\frac{1}{3}\cdot3t=\frac{1}{3}\cdot(\operatorname{arctg}5+\pi n)\)
\(t=\frac{1}{3}\cdot\operatorname{arctg}5+\frac{1}{3}\cdot\pi n\)
\(t=\frac{1}{3}\cdot\operatorname{arctg}5+\frac{\pi n}{3}\).
Ҷавоб: \(t_1=\frac{\pi}{12}(4n-1)\),
\(t_2=\frac{1}{3}\operatorname{arctg}5+\frac{\pi n}{3}\),\(\quad n\in \mathbb{Z}\).