Angle of Sun's RaysDate: 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/ |
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