December 25th, 2011, 11:18 PM  #1 
Senior Member Joined: Jul 2011 Posts: 400 Thanks: 15  trigonometric value
Without using a trigonometric table, calculate the value of: 
December 26th, 2011, 01:23 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
Using producttosum identities, cos(60Â°) = 1/2 and cos(36Â°)  sin(18Â°) = 1/2 (which I proved here), sin(12Â°)sin(48Â°)sin(54Â°) = (1/2)(cos(36Â°)  1/2)sin(54Â°) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ = (1/4)(2cos(36Â°)sin(54Â°)  sin(54Â°)) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ = (1/4)(1 + sin(18Â°)  sin(54Â°)) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ = (1/4)(1  (cos(36Â°)  sin(18Â°))) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ = (1/4)(1  1/2) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ = 1/8. Last edited by skipjack; October 18th, 2015 at 03:16 AM. 
December 26th, 2011, 06:50 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  Re: trigonometric value
First, a bit of advice for your ... For the degree symbol, use ^{\circ} and for the dot multiplication symbol use \cdot. Then use \( and \) to enclose expressions that are taller than one character, such as variables with exponents, or fractions to automatically get the right sized bracketing symbols. This works for any bracketing symbols. I edited your post to include these. Now to the problem, and much less elegantly than [color=#00BF00]skipjack[/color]: Using a producttosum identity and cos(x) = cos(x) we have: Using sin(90°  x) = cos(x), we have , thus: Using a tripleangle identity for sine on the left and a doubleangle identity for cosine on the right, we have: Factoring yields: Knowing we find: Using a doubleangle identity for cosine, we find: Putting it all together, we find: 
December 27th, 2011, 11:29 AM  #4 
Member Joined: Aug 2011 From: student Posts: 55 Thanks: 0  Re: trigonometric value Using producttosum: Using triple angle identities in sin(54)=sin(3Ã—18): Factorize: We have: :mrgreen: Last edited by skipjack; October 18th, 2015 at 03:07 AM. 
December 27th, 2011, 11:43 AM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  Re: trigonometric value
How did you know: ? 
December 28th, 2011, 01:13 AM  #6 
Member Joined: Aug 2011 From: student Posts: 55 Thanks: 0  Re: trigonometric value
Yes. Let's see: We have: Thus we have : Last edited by skipjack; February 25th, 2016 at 04:28 AM. 
December 28th, 2011, 10:57 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
That proof is equivalent to one that appears in the post I linked to above.


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