Question

Sa se rezolve : sin ( pi sin x ) + cos ( pi sin x ) = 1 Va rooog frumos !!!

Answer 1

   
[tex]\displaystyle\\ \sin(\pi \cdot \sin x)+\cos(\pi \cdot \sin x)=1\\ \text{Facem substitutia: } ~~~\boxed{(\pi \cdot \sin x)=2y}\\ \text{Rezulta ecuatia: }\\ \sin(2y)+\cos(2y)=1\\\\ \sin(2y)= \frac{2 \text{ tg }y }{1+\text{tg}^2y} \\\\ \cos(2y)= \frac{1-\text{tg}^2y}{1+\text{tg}^2y} \\\\ \text{Rezulta ecuatia:}\\\\ \frac{2 \text{ tg }y }{1+\text{tg}^2y}+ \frac{1-\text{tg}^2y}{1+\text{tg}^2y}=1\\\\ \frac{2 \text{ tg }y +1-\text{tg}^2y}{1+\text{tg}^2y}=1\\\\ 2 \text{ tg }y +1-\text{tg}^2y=1+\text{tg}^2y [/tex]

[tex]\displaystyle\\ -\text{tg}^2y-\text{tg}^2y +2 \text{ tg }y +1-1=0\\ -2\text{tg}^2y +2 \text{ tg }y=0~~~~~\Big|:(-2) \\ \text{tg}^2y -\text{tg }y=0\\ \text{tg }y(\text{tg }y-1)=0\\ \text{Rezulta 2 solutii:}\\ \text{tg }y=0 \Longrightarrow~~y=\boxed{0}\\ \text{tg }y-1 \Longrightarrow~~\text{tg }y=1 \Longrightarrow~~y= \boxed{\frac{\pi}{4}}\\ \text{Atentie! Nu am scrila solutii }+k\pi\text{ sau }+2k\pi \text{ etc, deoarece}\\ \text{asta nu stabileste tangenta, stabileste functia initiala.}\\\\ [/tex]


[tex]\displaystyle\\ \text{Revenim la ecuatia: }\\ \sin(2y)+\cos(2y)=1~~\text{unde}~~y_1=0~~\text{si}~~y_2=\frac{\pi}{4}\\\\ 2y = 2\cdot0=\boxed{0}~~\text{si}~~2y=2\cdot\frac{\pi}{4}=\boxed{\frac{\pi}{2}}\\\\ \text{Revenim la substitutia }~\boxed{(\pi\cdot\sin x)=2y}\\\\ \text{Solutia 1:}\\ \pi\cdot\sin x)=0\\ \sin x=0\\ \boxed{x_1=0+k\pi;~~k\in N} [/tex]

[tex]\displaystyle\\ \text{Solutia 2:}\\ \pi\cdot\sin x)=\frac{\pi}{2}\\\\ \sin x=\frac{\dfrac{\pi}{2}}{\pi}=\frac{\pi}{2\pi}=\frac{1}{2}\\\\ \boxed{x_2=\frac{\pi}{6}+2k\pi~~\bigcup~~ \frac{5\pi}{6}+2k\pi;~~k\in N}[/tex]