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June 8th, 2014, 10:18 PM   #1
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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
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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
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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
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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
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Quote:
Originally Posted by skipjack View Post
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?
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June 10th, 2014, 02:08 AM   #6
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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
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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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