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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 $\displaystyle \frac{\cos 12 ^\circ+\sin 36 ^\circ}{\cos 36^\circ-\sin 12 ^\circ}$. |
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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)) |
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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? ![]() |
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June 9th, 2014, 01:44 PM | #4 |
Global Moderator Joined: Dec 2006 Posts: 20,310 Thanks: 1978 |
Apply the trigonometric sum-to-product identities. $\displaystyle \frac{\cos(12°) - \cos(126°)}{\sin(126°) - \sin(12°)} = \frac{2\sin(69°)\sin(57°)}{2\cos(69°)\sin(57°) } = \tan(69°)$ |
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June 9th, 2014, 10:28 PM | #5 |
Newbie Joined: Jun 2014 From: USA Posts: 21 Thanks: 2 | |
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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. |
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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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