MHB Is Sin 1 Less Than Log Base 3 of Root 7?

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    2016
AI Thread Summary
The discussion centers on proving the inequality $\sin 1 < \log_3 \sqrt{7}$. Participants engage in mathematical reasoning to establish the validity of this statement. Kaliprasad successfully provides a correct solution, which is highlighted in the thread. The problem encourages deeper exploration of trigonometric and logarithmic functions. Overall, the thread emphasizes the importance of mathematical proof in understanding inequalities.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

Prove $\sin 1 < \log_3 \sqrt{7}$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
Congratulations to kaliprasad for his correct solution, which you can find below::)
We have $\sin \,1 < \sin \frac{\pi}{3} $ as $ 1 < \frac{\pi}{3}$
or $\sin \,1 < \frac{\sqrt{3}}{2}\cdots (1)$

Note that $(4\sqrt{3})^2 = 48 < 49 = 7^2$, hence $\sqrt{3} < \frac{7}{4}$

Combining the results we get
$\sin \,1 < \frac{7}{8}\cdots (2)$

Now observe that $3^7 = 2187$ and $7^4 = 2401$ hence

$3^7 < 7^4$

$3^\frac{7}{8} < \sqrt{7}$

$\frac{7}{8} < \log_3\sqrt{7}$

from (2) and above we have proved that $\sin\,1 < log_3 \sqrt{7}$.
 
Back
Top