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: 224 Thanks: 26 
$\displaystyle 3^{\sin2x}=2$ $\displaystyle 3^{\sin2x}=2=3^{\log_3 2}$ $\displaystyle \; \;$ $\displaystyle \;$ $\displaystyle \sin2x=\log_3 2$ $\displaystyle 2x=\arcsin (\log_3 2)$ $\displaystyle x=\frac{1}{2} \arcsin(\log_3 2)$ 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: 1,931 Thanks: 999 
$3^{\sin(2x)}=2$ $\sin(2x) = \log_3(2)$ $\sin(\pi  2x) = \log_3(2)$ 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: 1,797 Thanks: 715 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: 7,305 Thanks: 2443 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: 18,954 Thanks: 1600  
February 2nd, 2017, 12:55 PM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866 
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 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How do I solve an equation with a sin2x?  EasyGoingPatrick  Trigonometry  8  July 4th, 2016 01:05 AM 
Trig identities cos(x+3x)=sin4x/sin2x  atishba  Trigonometry  1  March 1st, 2016 04:24 PM 
integrate sin2x cos4x  xl5899  Calculus  4  December 22nd, 2015 04:46 AM 
min value y= sin2x+cosx  mared  PreCalculus  1  May 7th, 2014 12:25 PM 
integral of sin2x/2sincos^2x  nephi39  Calculus  6  January 1st, 2012 05:34 PM 