1. ## Trig

Show $\displaystyle \displaystyle \tan \frac\pi9 + 4\sin \frac\pi9 = \sqrt3$.

2. Originally Posted by chiph588@
Show $\displaystyle \displaystyle \tan \frac\pi9 + 4\sin \frac\pi9 = \sqrt3$.

Consider $\displaystyle \frac{\pi}{3} - \frac{\pi}{9} = \frac{2\pi}{9}$

$\displaystyle \sin(\frac{\pi}{3} - \frac{\pi}{9}) = \sin(\frac{2\pi}{9})$

$\displaystyle \frac{\sqrt{3}}{{2}} \cos(\frac{\pi}{9}) - \frac{1}{2} \sin(\frac{\pi}{9}) = 2\sin(\frac{\pi}{9}) \cos(\frac{\pi}{9})$

$\displaystyle \sqrt{3} \cos(\frac{\pi}{9}) - \sin(\frac{\pi}{9}) = 4 \sin(\frac{\pi}{9}) \cos(\frac{\pi}{9})$

$\displaystyle \sqrt{3} = \tan(\frac{\pi}{9}) + 4 \sin(\frac{\pi}{9})$

3. Originally Posted by simplependulum
Consider $\displaystyle \frac{\pi}{3} - \frac{\pi}{9} = \frac{2\pi}{9}$

$\displaystyle \sin(\frac{\pi}{3} - \frac{\pi}{9}) = \sin(\frac{2\pi}{9})$

$\displaystyle \frac{\sqrt{3}}{{2}} \cos(\frac{\pi}{9}) - \frac{1}{2} \sin(\frac{\pi}{9}) = 2\sin(\frac{\pi}{9}) \cos(\frac{\pi}{9})$

$\displaystyle \sqrt{3} \cos(\frac{\pi}{9}) - \sin(\frac{\pi}{9}) = 4 \sin(\frac{\pi}{9}) \cos(\frac{\pi}{9})$

$\displaystyle \sqrt{3} = \tan(\frac{\pi}{9}) + 4 \sin(\frac{\pi}{9})$
Much more elegant than my solution!

4. Originally Posted by chiph588@
Much more elegant than my solution!
What was your solution? De Moivre/binomial/Vieta?

5. Originally Posted by TheCoffeeMachine
What was your solution? De Moivre/binomial/Vieta?
Let $\displaystyle u = \cos\left(\frac\pi9\right)$. From the third angle formula, we obtain

$\displaystyle 8u^3-6u-1 = 0$

$\displaystyle -(2u+1)(8u^3-6u-1) = 0$

$\displaystyle -16u^4-8u^3+12u^2+8u+1 = 0$

$\displaystyle (1-u^2)(1+4u)^2 = 3u^2$

$\displaystyle \sqrt{1-u^2}(1+4u) = \sqrt3u$

$\displaystyle \sqrt{1-u^2}+4u\sqrt{1-u^2} = \sqrt3u$

$\displaystyle \sin\left(\frac\pi9\right)+4\cos\left(\frac\pi9\ri ght)\sin\left(\frac\pi9\right) = \sqrt3\cos\left(\frac\pi9\right)$

$\displaystyle \tan\left(\frac\pi9\right)+4\sin\left(\frac\pi9\ri ght) = \sqrt3$

6. Originally Posted by chiph588@
Let $\displaystyle u = \cos\left(\frac\pi9\right)$. From the third angle formula, we obtain

$\displaystyle 8u^3-6u-1 = 0$

$\displaystyle -(2u+1)(8u^3-6u-1) = 0$

$\displaystyle -16u^4-8u^3+12u^2+8u+1 = 0$

$\displaystyle (1-u^2)(1+4u)^2 = 3u^2$

$\displaystyle \sqrt{1-u^2}(1+4u) = \sqrt3u$

$\displaystyle \sqrt{1-u^2}+4u\sqrt{1-u^2} = \sqrt3u$

$\displaystyle \sin\left(\frac\pi9\right)+4\cos\left(\frac\pi9\ri ght)\sin\left(\frac\pi9\right) = \sqrt3\cos\left(\frac\pi9\right)$

$\displaystyle \tan\left(\frac\pi9\right)+4\sin\left(\frac\pi9\ri ght) = \sqrt3$
If you're wondering how I came up with this, it's called working backwards .