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June 2nd, 2019, 06:49 AM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 651 Thanks: 86  Quick Geometry and Trigonometry Questions
1. If a right triangle has equal legs, the hypotenuse divided by a leg is the square root of 2. If a right triangle has unequal legs, is it always true that the hypotenuse divided by the longer leg is less than the square root of 2 and the hypotenuse divided by the shorter leg is greater than the square root of 2? 2. A rational angle can produce rational or irrational trigonometry values. If a trigonometry value is rational, does that guarantee it came from a rational angle? By trigonometry value I mean the sine, cosine, or tangent of any angle. 
June 2nd, 2019, 08:17 AM  #2  
Math Team Joined: Jul 2011 From: Texas Posts: 2,982 Thanks: 1575  Quote:
case 1 ... $\dfrac{c^2}{a^2} = 1 + \dfrac{b^2}{a^2}$ $\dfrac{c}{a} = \sqrt{1 + \dfrac{b^2}{a^2}} < \sqrt{1 + 1}$ case 2 ... $\dfrac{c^2}{b^2} = 1 + \dfrac{a^2}{b^2}$ $\dfrac{c}{b} = \sqrt{1 + \dfrac{a^2}{b^2}} > \sqrt{1 + 1}$  
June 2nd, 2019, 09:07 AM  #3  
Math Team Joined: Jul 2011 From: Texas Posts: 2,982 Thanks: 1575  Quote:
https://math.stackexchange.com/quest.../299138#299138  
June 2nd, 2019, 02:27 PM  #4  
Senior Member Joined: Oct 2013 From: New York, USA Posts: 651 Thanks: 86  Quote:
Quote:  
June 2nd, 2019, 02:36 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,835 Thanks: 2162 
There is no rational angle whose sine is 1/2.

June 2nd, 2019, 05:39 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
$\displaystyle \sin\left(30^\circ\right)=\frac12$


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