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January 31st, 2017, 12:16 PM  #1 
Senior Member Joined: Apr 2008 Posts: 174 Thanks: 3  simplify without double angle identities
I am going to write down a math problem below. (ex) Without using double angle identities, simplify 2cos(3x)*sinx my attempt: 2cos(3x)*sinx =2cos(x+2x)*sinx =2sinx*cosx*cos(2x)2sinx*sinx*sin(2x) =2sinx*cosx*cos(2x)2(sinx)^2*sin(2x) =2sinx*cosx*(cosx*cosxsinx*sinx)2sin(2x)+2(cosx)^2*sin(2x) =2sinx*(cosx)^32(sinx)^3*cosx2sin(2x)+2(cosx)^2*sin(2x) My strategy makes the original expression very messy. I am really stuck. The answer is sin(4x)sin(2x). Can someone explain how to do the problem without double angle identities? Thanks a lot. 
January 31st, 2017, 01:27 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,488 Thanks: 887 Math Focus: Elementary mathematics and beyond 
$$2\cos(3x)\sin(x)=\sin(3x+x)\sin(3xx)=\sin(4x)\sin(2x)$$

January 31st, 2017, 02:52 PM  #3 
Senior Member Joined: Apr 2008 Posts: 174 Thanks: 3 
Thanks. greg1313. There is something I don't get. What made you use sin(3x+x)  sin(3xx)? This is not obvious to me.

January 31st, 2017, 04:26 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,488 Thanks: 887 Math Focus: Elementary mathematics and beyond 
The problem you posted is a specific case of a well known identity.

January 31st, 2017, 07:02 PM  #5 
Member Joined: Oct 2016 From: Melbourne Posts: 77 Thanks: 35  
January 31st, 2017, 11:57 PM  #6 
Senior Member Joined: Apr 2008 Posts: 174 Thanks: 3 
Because I didn't know anything about this wellknown identity, I had a hard time with the problem. How would you approach it if you didn't know that identity?

February 1st, 2017, 02:08 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 17,166 Thanks: 1285 
Alternatively, use $\cos(\theta) = (e^{i\theta} + e^{i\theta})/2$, $\sin(\theta) = (e^{i\theta}  e^{i\theta})/(2i)$, and the law of exponents. The task then reduces to easy algebra.

February 2nd, 2017, 10:27 PM  #8 
Senior Member Joined: Apr 2008 Posts: 174 Thanks: 3 
This problem "looks" easy, but it requires you to use advanced ideas to solve it like the polar form of sine and cosine. I wish there was a much easier way to do it. Thanks, everyone. 

Tags 
angle, double, identities, simplify 
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