Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
[tex]2cos^2x+sinx-1=0\\\\2(1-sin^2x)+sinx-1=0\\\\2-2sin^2x+sinx-1=0\\\\-2sin^2x+sinx+1=0\\\\-2sin^2x+2sinx-sinx+1=0\\\\-2sinx(sinx-1)-1(sinx-1)=0\\\\(sinx-1)(-2sinx-1)=0\iff sinx-1=0\ or\ -2sinx-1=0\\\\sinx=1\ or\ -2sinx=1\\\\sinx=1\ or\ sinx=-\frac{1}{2}\\\\x=\frac{\pi}{2}+2k\pi\ or\ x=-\frac{\pi}{6}+2k\pi\ or\ x=\frac{7\pi}{6}+2k\pi\ where\ k\in\mathbb{Z}[/tex]
[tex]2cosx^2+sinx-1=2(1-sin^2x)+snx-1=\\\\=2(1-sinx)(1+sinx)-(1-sinx)=(1-sinx)[2(1+sinx)-1]=\\\\=(1-sinx)(2+2sinx-1)=(1-sinx)(1+2sinx)\\\\2cosx^2+sinx-1=0\ \ \ \Leftrightarrow\ \ \ (1-sinx)(1+2sinx)=0\\\\1-sinx=0\ \ \ \ \ or\ \ \ \ \ 1+2sinx=0\\\\1)\ \ \ 1-sinx=0\ \ \ \Rightarrow\ \ \ sinx=1\ \ \ \Rightarrow\ \ \ x= \frac{ \pi }{2} +2k \pi ,\ \ \ k\in I\\\\[/tex]
[tex]2)\ \ \ 1+2sinx=0\ \ \ \ \ \ \Rightarrow\ \ \ sinx=- \frac{1}{2}\\\\ \Rightarrow\ \ \ x_1=( \pi + \frac{ \pi }{6} )+2k \pi ,\ \ \ \ \ \ x_2=( - \frac{ \pi }{6} )+2k \pi,\ \ \ \ \ \ k\in I\\\\.\ \ \ \ \ \ x_1=\frac{7 \pi }{6} +2k \pi ,\ \ \ \ \ \ \ \ \ \ \ \ x_2=-\frac{ \pi }{6} +2k \pi,\ \ \ \ \ \ \ \ \ k\in I\\\\Ans.\ x=-\frac{ \pi }{6} +2k \pi\ \ \ or\ \ \ x= \frac{ \pi }{2} +2k \pi\ \ \ or\ \ \ x=\frac{7 \pi }{6} +2k \pi,\ \ \ k\in I[/tex]
[tex]2)\ \ \ 1+2sinx=0\ \ \ \ \ \ \Rightarrow\ \ \ sinx=- \frac{1}{2}\\\\ \Rightarrow\ \ \ x_1=( \pi + \frac{ \pi }{6} )+2k \pi ,\ \ \ \ \ \ x_2=( - \frac{ \pi }{6} )+2k \pi,\ \ \ \ \ \ k\in I\\\\.\ \ \ \ \ \ x_1=\frac{7 \pi }{6} +2k \pi ,\ \ \ \ \ \ \ \ \ \ \ \ x_2=-\frac{ \pi }{6} +2k \pi,\ \ \ \ \ \ \ \ \ k\in I\\\\Ans.\ x=-\frac{ \pi }{6} +2k \pi\ \ \ or\ \ \ x= \frac{ \pi }{2} +2k \pi\ \ \ or\ \ \ x=\frac{7 \pi }{6} +2k \pi,\ \ \ k\in I[/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.