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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
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Quote:
 Originally Posted by EvanJ 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?
$a^2+b^2 = c^2$ with $a > b$

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
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Quote:
 Originally Posted by EvanJ 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.

https://math.stackexchange.com/quest.../299138#299138

June 2nd, 2019, 02:27 PM   #4
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Quote:
 Originally Posted by skeeter $a^2+b^2 = c^2$ with $a > b$ 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 ... $\dfrac{c^2}{b^2} = 1 + \dfrac{a^2}{b^2}$ $\dfrac{c}{b} = \sqrt{1 + \dfrac{a^2}{b^2}} > \sqrt{1 + 1}$
Thank you.

Quote:
That's about if the trig functions are rational. I'm asking if a rational trig function guarantees that it came from a rational angle measure.

 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. Thanks from EvanJ
 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 $\displaystyle \sin\left(30^\circ\right)=\frac12$ Thanks from topsquark

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