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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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 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:
 Originally Posted by skeeter link within a link ... https://math.stackexchange.com/quest.../299138#299138
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 Tags geometry, questions, quick, trigonometry Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Math_genius88 Geometry 6 August 2nd, 2010 04:43 AM bignick79 Algebra 2 June 15th, 2010 03:40 PM football Algebra 2 May 16th, 2010 09:47 PM packerfan128 Calculus 4 December 9th, 2008 03:05 PM packerfan128 Calculus 5 September 22nd, 2008 06:18 PM

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