January 25th, 2017, 01:03 PM  #1 
Newbie Joined: Jan 2017 From: Canada Posts: 2 Thanks: 0  3^sin2x=2
This was a test question I don't know how to do it, if you can help me please!

January 25th, 2017, 01:36 PM  #2 
Senior Member Joined: Dec 2015 From: Earth Posts: 248 Thanks: 27 
$\displaystyle 3^{\sin2x}=2$ $\displaystyle 3^{\sin2x}=2=3^{\log_3 2}$ $\displaystyle \; \;$ $\displaystyle \;$ $\displaystyle \sin2x=\log_3 2$ $\displaystyle 2x=\arcsin (\log_3 2)$ $\displaystyle x=\frac{1}{2} \arcsin(\log_3 2)$ Last edited by skipjack; January 25th, 2017 at 07:13 PM. 
January 25th, 2017, 01:42 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 
$3^{\sin(2x)}=2$ $\sin(2x) = \log_3(2)$ $\sin(\pi  2x) = \log_3(2)$ as well  $\sin(2x) = \log_3(2)$ $\sin(2x + 2k\pi) = \log_3(2),~k \in \mathbb{Z}$ $2(x+k \pi) = \arcsin(\log_3(2))$ $x = \dfrac{\arcsin(\log_3(2))}{2}k \pi,~k \in \mathbb{Z}$  $\sin(\pi  2x) = \log_3(2)$ $\sin(\pi  2x+ 2k\pi) = \log_3(2),~k \in \mathbb{Z}$ $\sin((2k+1) \pi  2x ) = \log_3(2),~k \in \mathbb{Z}$ $(2k+1)\pi  2x = \arcsin(\log_3(2))$ $2x = \arcsin(\log_3(2))(2k+1)\pi$ $x = \dfrac{(2k+1)\pi  \arcsin(\log_3(2))}{2}$ so the final result is $x \in \left\{\dfrac{\arcsin(\log_3(2))}{2}k \pi\right \} \cup \left\{\dfrac{(2k+1)\pi  \arcsin(\log_3(2))}{2}\right\},~k \in \mathbb{Z}$ 
January 25th, 2017, 01:43 PM  #4 
Newbie Joined: Jan 2017 From: Canada Posts: 2 Thanks: 0 
I don't know what ark is. I'm only in advance functions. Maybe do you know another way, besides using ark? Last edited by Abhikh; January 25th, 2017 at 01:54 PM. 
January 25th, 2017, 03:37 PM  #5  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,889 Thanks: 769 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Frankly, I prefer to use asn(x) but that is a notation that isn't used much any more, so when I'm on the forums I use $\displaystyle \sin^{1}(x)$. Dan Last edited by skipjack; January 25th, 2017 at 07:15 PM.  
January 25th, 2017, 05:45 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
I avoid $\sin^{1}x$ as it easily creates confusion between $\arcsin x$ and $ \frac{1}{\sin x}$, especially if you wish to write $\sin^2 x = (\sin x)^2$.

January 25th, 2017, 07:19 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 19,713 Thanks: 1806  
February 2nd, 2017, 12:55 PM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
I believe that skipjack's point is that "3^sin(2x)= 2" is not a "question", it is an equation. What were you asked to do with that equation, solve it for x?


Tags 
3sin2x2, log, sin 
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