Angle of Sun's Rays

Date: 05/03/99 at 17:18:26
From: Ken Mullan
Subject: Determining angles

How could you determine the angle at which the sun's light hits the 
earth at any given point? (The two lines that form the angle are the 
surface of the earth, and the sun ray.) I have no idea how to figure 
it out. Thank you!

Date: 05/05/99 at 12:22:17
From: Doctor Jeff
Subject: Re: Determining angles

Hello, Ken. 

To answer this question, you need to know a little trigonometry. 
Specifically, in the right triangle below, the tangent of angle A is 
defined as y/x.

For example, if angle A were equal to 60 degrees, then

     tanA = y/x = tan(60) = sqrt(3)/1 = sqrt(3)
              .
             /| 
            / |
           /  |
          /   |
         /    |y
        /     |
       /      |
      /       |
   A /________|
          x

Here are a couple other common tangents:

     tan(30) = sqrt(3)/3
     tan(45) = 1/1 = 1

You can find an approximation for tanA when angle A is unfamiliar by 
using a scientific or graphing calculator.

Now we're ready to look at your problem.
                    _
                   |_|
                   sun
                  /
                 /
                /
               /
              /
             /| 
      light / |
       ray /  |
          /   |object
         /    |of known
        /     |height
       /      |
      /       |
   A /________|
      shadow
      of object

The best way to measure angle A, the angle made by the light ray and 
the earth, is to place an object of known height - perhaps a yardstick 
- perpendicular to the ground. Then measure the length of its shadow. 
As you can see by the picture,

                length of object
     tanA = -------------------------
            length of object's shadow


One method for finding A is to hope that tanA is one of the common 
ones mentioned earlier. For instance, if the length of the object is 3 
and the length of its shadow is 3, then

     tanA = 3/3 = 1/1 = 1

We already know that the angle that gives us a tangent of 1 is 45 
degrees, so we just found A. Most of the time, however, you will have 
to use a calculator to determine the angle A. For example, if the 
object's length is 8 and its shadow is 5, we get

     tanA = 8/5

We need to introduce the term "arctangent." For

     tanA = y/x,
     arctan(y/x) = A, so
     A = arctan(y/x)

So, in the above example, to find A, we just take arctan(8/5). Using 
a calculator, we find that A is approximately equal to 58 degrees. If 
you are having a hard time finding the arctangent button on your 
calculator, it probably looks like 

        -1
     TAN

Hope this explanation helped. If you need clarification, please write 
back!

- Doctor Jeff, The Math Forum
  http://mathforum.org/dr.math/