Angle of Elevation

Date: 01/22/97 at 21:11:59
From: David
Subject: I need some help in the Geometry Area 

I working on angle of elevation and angle of depression right now and 
I have a question about this problem:

A tree 66 meters high casts a 44 meter shadow.  Find the measure of 
the angle of elevation of the sun.

We are using cosine, sine, and tangent right now, so how do I do this 
problem using this information? 

Sincerely, David

Date: 01/22/97 at 21:51:44
From: Doctor Wallace
Subject: Re: I need some help in the Geometry Area 

Hi David!

Okay, the first thing to do in a problem of this sort is draw a 
picture.  I hope you did that.

I drew a line on the right side of my paper representing the tree.  I 
labeled it 66m.  The sun is higher than the tree, and to the right of 
it.  I drew a straight line coming from the sun diagonally down to the 
left, touching the top of the tree and continuing on for some 
distance.  This is the sunbeam.  Now for the shadow.  I drew a line 
perpendicular to the tree, from the bottom of the three to the left 
until it intersected the sunbeam.  This line I labeled 44m.

You notice that what you have drawn is a right triangle.  The right 
angle is in the lower right, and you have 2 acute angles, one at the 
top right, and one at the bottom left.  Okay so far?

Now, the problem asks you for the angle of elevation of the sun.  Do 
you know which of the triangle's angles this is?  Pretend you are 
standing on the ground to the left of the tree, at the end of the 
shadow.  You look up at the sun.  The angle of elevation of the sun is 
how high the sun is above the ground from that point.  So the lower 
left angle of the right triangle is the angle you're going for.  

Now, how to find it?  Well, you mentioned that you need sin, cos, and 
tan. Good job. But which one?  Well, sine is opposite over hypotenuse, 
cosine is adjacent over hypotenuse, and tangent is opposite over 
adjacent.

You should be able to take it from here.  Only one of these three 
trigonometric functions can be applied here (if you look at your 
triangle, you'll see that you don't know the length of the 
hypotenuse).  Then you will have an equation like:
 
angle of elevation = something divided by something

Then you can either use a calculator or the table of trig functions in 
your math book to find the angle.

-Doctor Wallace,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/