June 8th, 2014, 09:18 PM  #1 
Senior Member Joined: Apr 2013 Posts: 425 Thanks: 24  An exact value
Hello! To calculate the exact value of the fraction $\displaystyle \frac{\cos 12 ^\circ+\sin 36 ^\circ}{\cos 36^\circ\sin 12 ^\circ}$. 
June 9th, 2014, 12:07 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,160 Thanks: 1638 
It simplifies to tan(69°). If you really want the exact value of that, write it as (tan(45°) + tan(24°))/(1  tan(45°)tan(24°)). tan(45°) = 1 and tan(24°) = √(23 + 10√5  2√(255 + 114√5)) 
June 9th, 2014, 04:45 AM  #3 
Member Joined: Feb 2012 From: Hastings, England Posts: 83 Thanks: 14 Math Focus: Problem Solving 
wow, sorry to keep questioning everyone, but where does the 69 come from? 
June 9th, 2014, 12:44 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,160 Thanks: 1638 
Apply the trigonometric sumtoproduct identities. $\displaystyle \frac{\cos(12°)  \cos(126°)}{\sin(126°)  \sin(12°)} = \frac{2\sin(69°)\sin(57°)}{2\cos(69°)\sin(57°) } = \tan(69°)$ 
June 9th, 2014, 09:28 PM  #5 
Newbie Joined: Jun 2014 From: USA Posts: 21 Thanks: 2  
June 10th, 2014, 01:08 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,160 Thanks: 1638 
The first fraction is equivalent to the fraction posted originally. For the standard trigonometric identities, read this article. 
June 13th, 2014, 11:20 PM  #7 
Banned Camp Joined: Feb 2013 Posts: 224 Thanks: 6  Illusion of Rationality
Those angles are Euclidean angles (multiples of 3 degrees). In the real world there are all kinds of angles. Natural angles such as 20 degrees. You just punch in a calculator. Or if you don't have a calculator, use a protractor to get analog distances.


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