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June 2nd, 2019, 06:49 AM   #1
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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.
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June 2nd, 2019, 08:17 AM   #2
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Quote:
Originally Posted by EvanJ View Post
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 ...

$\dfrac{c^2}{b^2} = 1 + \dfrac{a^2}{b^2}$

$\dfrac{c}{b} = \sqrt{1 + \dfrac{a^2}{b^2}} > \sqrt{1 + 1}$
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June 2nd, 2019, 09:07 AM   #3
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Originally Posted by EvanJ View Post
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.
link within a link ...

https://math.stackexchange.com/quest.../299138#299138
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June 2nd, 2019, 02:27 PM   #4
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Quote:
Originally Posted by skeeter View Post
$a^2+b^2 = c^2$ 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 ...

$\dfrac{c^2}{b^2} = 1 + \dfrac{a^2}{b^2}$

$\dfrac{c}{b} = \sqrt{1 + \dfrac{a^2}{b^2}} > \sqrt{1 + 1}$
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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.
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June 2nd, 2019, 02:36 PM   #5
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There is no rational angle whose sine is 1/2.
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June 2nd, 2019, 05:39 PM   #6
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$\displaystyle \sin\left(30^\circ\right)=\frac12$
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