# prove 2

• Jul 21st 2007, 03:03 AM
perash
prove 2
tan 20 + 4sin 20 = sqrt 3
• Jul 21st 2007, 04:06 AM
DivideBy0
Quote:

Originally Posted by perash
tan 20 + 4sin 20 = sqrt 3

They are not equal.
• Jul 21st 2007, 04:50 AM
topsquark
Quote:

Originally Posted by perash
tan 20 + 4sin 20 = sqrt 3

Quote:

Originally Posted by DivideBy0
They are not equal.

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

-Dan
• Jul 21st 2007, 04:52 AM
DivideBy0
O geeze :o it seems i used radians, so sorry
• Jul 21st 2007, 09:24 AM
red_dog
The equality is equivalent with
$\sin 20+4\sin 20\cos 20=\sqrt{3}\cos 20\Leftrightarrow$
$\Leftrightarrow \sin 20+2\sin 40=\sqrt{3}\cos 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\frac{\sqrt{3}}{2}\cos 20-\frac{1}{2}\sin 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\cos 30\cos 20-\sin 30\sin 20\Leftrightarrow$
$\Leftrightarrow \sin 40=\cos 50\Leftrightarrow$
$\Leftrightarrow \sin 40=\sin 40$
• Jul 21st 2007, 09:33 AM
topsquark
Quote:

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
• Jul 21st 2007, 09:57 AM
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.
• Jul 21st 2007, 10:16 AM
topsquark
Quote:

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