$\sin 9^{\circ}$ or $\tan 8^{\circ} $ which one is bigger? $\sin 9^{\circ}$ or $\tan 8^{\circ} $ which one is bigger ?
someone ask me that , and said without using calculator !!
now my question is ,how to find which is bigger ?
  Is there a logical way to find  ?
I s there a mathematical method to show which is greater ?
I am thankful for your guide , hint  or solution 

I would check the first two terms of the Taylor series.  $\sin 9^\circ \approx \frac \pi{20}-\frac {\pi^3}{6 \cdot 20^3}, \tan 8^\circ \approx \frac {2\pi}{45}+\frac {8\pi^3}{3 \cdot 45^3}$, so $$\sin 9^\circ -\tan 8^\circ\approx \frac \pi{20}-\frac {\pi^3}{6 \cdot 20^3}-\frac {2\pi}{45}-\frac {8\pi^3}{3 \cdot 45^3}\\\approx \frac \pi{180}-\frac{(3^6+2^{10})\pi^3}{2^73^75^3}\\
 \approx \frac \pi{180}(1-\frac {1753}{2^43^55}) \\ \approx \frac \pi{180}(1-\frac {1753}{17440})\\ \gt 0$$
where I used $\pi^2 \approx 10$.  Alpha agrees, but I didn't check until I was done.
