An exact value

Dacu · June 8th, 2014, 10:18 PM

Hello!
To calculate the exact value of the fraction $\displaystyle \frac{\cos 12 ^\circ+\sin 36 ^\circ}{\cos 36^\circ-\sin 12 ^\circ}$.

skipjack · June 9th, 2014, 01:07 AM

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))

nmenumber1 · June 9th, 2014, 05:45 AM

wow, sorry to keep questioning everyone, but where does the 69 come from?

skipjack · June 9th, 2014, 01:44 PM

Apply the trigonometric sum-to-product identities.

$\displaystyle \frac{\cos(12°) - \cos(126°)}{\sin(126°) - \sin(12°)} = \frac{2\sin(69°)\sin(57°)}{2\cos(69°)\sin(57°) } = \tan(69°)$

peterwilliam · June 9th, 2014, 10:28 PM

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?

skipjack · June 10th, 2014, 02:08 AM

The first fraction is equivalent to the fraction posted originally.

For the standard trigonometric identities, read this article.

long_quach · June 14th, 2014, 12:20 AM

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.

June 8th, 2014, 10:18 PM	#1
Dacu Senior Member Joined: Apr 2013 Posts: 425 Thanks: 24	An exact value Hello! To calculate the exact value of the fraction $\displaystyle \frac{\cos 12 ^\circ+\sin 36 ^\circ}{\cos 36^\circ-\sin 12 ^\circ}$.
$Dacu is offline$

June 9th, 2014, 01:07 AM	#2
skipjack 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))
$skipjack is offline$

June 9th, 2014, 05:45 AM	#3
nmenumber1 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?
$nmenumber1 is offline$

June 9th, 2014, 01:44 PM	#4
skipjack Global Moderator Joined: Dec 2006 Posts: 20,310 Thanks: 1978	Apply the trigonometric sum-to-product identities. $\displaystyle \frac{\cos(12°) - \cos(126°)}{\sin(126°) - \sin(12°)} = \frac{2\sin(69°)\sin(57°)}{2\cos(69°)\sin(57°) } = \tan(69°)$
$skipjack is offline$

June 10th, 2014, 02:08 AM	#6
skipjack 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.
$skipjack is offline$

June 14th, 2014, 12:20 AM	#7
long_quach 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.
$long_quach is offline$

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how to find exact value of tan69