Expert: Azeem Hussain - 10/8/2009

Question
QUESTION: Hey,
I'm a high school junior and I'm trying to figure out how to find the elevation of the sun by using the shadow of a meter stick and a protractor. My textbook has this certain method for doing it but I don't get how I can possibly use the protractor to measure the shadow's distance from the sun.
 Help!

Oh and by the way, we haven't learned anything about inverse tangents(I was Googling and found that a lot of people used this to figure it out), so there HAS to be some way to solve it without using tangents.

Thanks in advance!

Bella S.



ANSWER: Hi Bella,

The method I will explain to you uses an inverse tangent.  Find some flat ground, and hold the metre stick perpendicular to it.  Make a mark on the ground where the stick's shadow ends.  (If you have the means, trace the shadow back to the metre stick.  Then lay the metre stick down, while maintaining the right angle with the trace of its shadow.  Connect the triangle on the ground.)  If you have a protractor, measure the angle from the tip of the shadow to the tip of thee metre stick.  This is the angle of elevation of the sun.  If you don't have a protractor, and even if you do it's probably more precise like this, divide the metre stick's length by its shadow's length, and take the inverse tangent of that.

If you have a protractor and you don't want to use an inverse tangent, then measure the angle directly.  I don't know of an accurate method that does not use an inverse tangent.  Until you learn about inverse trigonometry, they're just buttons on your calculator. Don't be afraid of them.  If you want, ask a follow-up and I'll explain the inverse tangent to you.

Thanks for asking,
Azeem

---------- FOLLOW-UP ----------

QUESTION: Hey,

Could you explain inverse tangent to me? We haven't learned anything about tangents yet, so it would be nice if you could give me a quick definition of that too.

Thank you, though, for your initial answer! It's very useful.

Bella

Answer
Hey Bella,

When dealing with a right triangle, basic trigonometry comes into play.  Draw a right triangle.  Label one of the acute angles <x.  Call the longest side the hypotenuse (hyp).  Call the side opposite <x opposite (opp).  Call the third side adjacent (adj).  Here are three fundamental relationships:
sin (<x)=opp/hyp
cos (<x)=adj/hyp
tan (<x)=opp/adj

A tangent (tan) is defined as the ratio of the opposite side to the adjacent side in a right triangle.  (A more general definition will be given later.)  Also, tan(<x) is not "tan times x," but rather "tan of x."  tan is an operator.

These relationships are useful for finding either a side or an angle.  If you want to find the angle, you need an arcsine, arrcosine, or arctangent, denoted arcsin, arccos, arctan.  A more common notation is tan^-1.  This means inverse tan or arctan, not "tan to the power of negative one."  (I prefer arctan because there is no ambiguity.)  Arctan is the inverse of tan, like the way multiplication is the inverse of division.  For instance, you would divide both sides by 2 in the following example to solve for x:
2x=10
2x/2=10/2
x=5

Similarly, you'd take the arctan of both sides to find the angle x.
tan x=opp/adj
arctan(tan x)=arctan(opp/adj)
x=arctan(opp/adj)

That's a quick intro to tangents as a trigonometric operator.  In general, tangent refers to a line that hits a curve at one point (simplified definition).  You probably won't see the connection between the geometric definition and the trigonometric definition in school.

Anyway, hope this helped!

Thanks for asking,
Azeem