# Math Help - prove 2

1. ## prove 2

tan 20 + 4sin 20 = sqrt 3

2. Originally Posted by perash
tan 20 + 4sin 20 = sqrt 3
They are not equal.

3. 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

4. O geeze it seems i used radians, so sorry

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

6. 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

7. 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.

8. 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