prove 2

**perash** · July 21st 2007, 02:03 AM

tan 20 + 4sin 20 = sqrt 3

**DivideBy0** · July 21st 2007, 03:06 AM

Originally Posted by perash

tan 20 + 4sin 20 = sqrt 3

They are not equal.

**topsquark** · July 21st 2007, 03:50 AM

Originally Posted by perash

tan 20 + 4sin 20 = sqrt 3

Originally Posted by DivideBy0

They are not equal.

If they aren't equal they are awfully darned close!

-Dan

**DivideBy0** · July 21st 2007, 03:52 AM

O geeze

it seems i used radians, so sorry

**red_dog** · July 21st 2007, 08:24 AM

The equality is equivalent with
$\sin 20+4\sin 20\cos 20=\sqrt{3}\cos 20\Leftrightarrow$
$\Leftrightarrow \sin 20+2\sin 40=\sqrt{3}\cos 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\frac{\sqrt{3}}{2}\cos 20-\frac{1}{2}\sin 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\cos 30\cos 20-\sin 30\sin 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\cos 50\Leftrightarrow$
$\Leftrightarrow \sin 40=\sin 40$

**topsquark** · July 21st 2007, 08:33 AM

Originally Posted by red_dog

The equality is equivalent with
$\sin 20+4\sin 20\cos 20=\sqrt{3}\cos 20\Leftrightarrow$
$\Leftrightarrow \sin 20+2\sin 40=\sqrt{3}\cos 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\frac{\sqrt{3}}{2}\cos 20-\frac{1}{2}\sin 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\cos 30\cos 20-\sin 30\sin 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\cos 50\Leftrightarrow$
$\Leftrightarrow \sin 40=\sin 40$

Daaaaaaaaaaaaaang! Do you have a process for figuring these things out or does it just take intuition and experience?

-Dan

**DivideBy0** · July 21st 2007, 08:57 AM

Awesome stuff red_dog, it looked all the more harder because I've heard that it is not possible to give an exact value for any angle which is not a multiple of 3.

**topsquark** · July 21st 2007, 09:16 AM

Originally Posted by DivideBy0

Awesome stuff red_dog, it looked all the more harder because I've heard that it is not possible to give an exact value for any angle which is not a multiple of 3.

Funny, that. I can get an exact answer for
$sin(22.5) = sin \left ( \frac{45}{2} \right )$
and 22.5 is not divisible by 3.

-Dan

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