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 June 8th, 2014, 10:18 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, 01:07 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,310 Thanks: 1978 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, 05:45 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, 01:44 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,310 Thanks: 1978 Apply the trigonometric sum-to-product 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, 10:28 PM   #5
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Quote:
 Originally Posted by skipjack Apply the trigonometric sum-to-product identities. $\displaystyle \frac{\cos(12°) - \cos(126°)}{\sin(126°) - \sin(12°)} = \frac{2\sin(69°)\sin(57°)}{2\cos(69°)\sin(57°) } = \tan(69°)$
How we solve with this formula?

 June 10th, 2014, 02:08 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,310 Thanks: 1978 The first fraction is equivalent to the fraction posted originally. For the standard trigonometric identities, read this article.
 June 14th, 2014, 12:20 AM #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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