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I am puzzled by a question in Trigonometry by Gelfand and Saul on p. 57.

Can it happen that an object will not cast any shadow at all? When and where? You may need to know something about astronomy to answer this question.

I have drawn a diagram with the height of the object represented by $h$ and the length of the shadow $l$ ( I don't know how to upload it, sorry).

To calculate the length of the shadow I used

$\cot \theta = \dfrac{l}{h}$

Which rearranging gives

$l = h\cot \theta$

We want $l = 0 $, which I think occurs when $\theta = 90$. I say think because my calculator says $tan 90$ is a "math error" (my calculator can't calculate $\cot$ directly). Am I correct in saying the shadow is of zero length when $\theta = 90$ ?

Secondly my astronomy is less than it could be. Where and when would the sun create an angle of 90 degrees? I am thinking at noon. Does this occur at any latitude?

mikoyan
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  • What is $\theta$? – Chris Eagle Jul 14 '13 at 13:37
  • Sorry it is the angle from the end of the shadow to the top of the object. – mikoyan Jul 14 '13 at 13:39
  • Whenever the sun will be exactly above (or 90 degrees above, as you said), then there will be no shadow at all. But it is only if you are standing on the line of equator. Since rays should act perpendicular to the earth or exactly parallel to you, and it is only if you are standing on the equator. So to answer your question, you better be headed to south America, Africa or indonisia.. – Rohinb97 Jul 14 '13 at 15:28
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    Actually, any point on earth in the Tropics (between the Tropic of Cancer and the Tropic of Capricorn) will experience a vertical sun twice each year – DJohnM Jul 14 '13 at 15:38
  • Does the sun cast a shadow? This question feels more like a brainteaser than a trigonometry question. At best, it's a quite open-ended trigonometry question. – Pete L. Clark Jul 14 '13 at 21:36

2 Answers2

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What is the object? A shallow spherical cap will only cast a shadow when the sun is near the horizon, although it can be held on its side to make it cast a shadow or lit from below. There are lots of objects like prime numbers and functors that never cast shadows.

$\displaystyle \cot(90)=\lim_{x\to 90^+} \frac{1}{\tan(x)}\lim_{x\to 90^-} \frac{1}{\tan(x)}=0$

At equinox the sun is at the zenith (perpendicularly overhead) at the point on the equator where it is noon. At any instant there is a point somewhere between the tropics of Cancer and Capricorn where the sun is at zenith.

Angela Pretorius
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  • The exact nature of the object is undefined in the question. I am assuming a stick. Thank you for the information on the tropics of Cancer and Capricorn and the confirmation that $cot(90) = 0)$. – mikoyan Jul 14 '13 at 14:25
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Unless your stick has zero thickness, the answer is no. Consider the sun from our earthly point of view. It obviously isn't a point source, but rather it looks like a disk (subtending an angle of approximately half a degree, though that's not important here). Place your stick so that it points upward exactly to the center of the sun, if possible. The light coming from the rim of the sun's apparent disk will not strike the top of the disk at exactly zero degrees (measured from vertical), but instead will cast a small faint shadow (subtending about half a degree, by similar triangles) all around the base of the stick. If I did the calculations correctly, that's about a 4 mm radius shadow for a 1 m stick.

Rick Decker
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