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Trig Ratios of a Right Angle in a Right Triangle

Date: 09/21/2006 at 15:41:54
From: Alex
Subject: What the opposite and adjacent sides of a 90 degree angle

There was a question asking to find the smallest tangent of an angle 
of a right triangle.  While I was doing this, I saw the 90 degree
angle and wondered what was the opposite and what was the adjacent
side.  Is the hypotenuse the opposite side?  Please help.

Date: 09/22/2006 at 08:18:06
From: Doctor Rick
Subject: Re: What the opposite and adjacent sides of a 90 degree angle

Hi, Alex.

In order to apply the opposite-adjacent-hypotenuse definitions of the 
trig functions, the angle whose trig functions we are considering must 
be one of the acute angles of a right triangle.  That way, there are 
TWO known angles of the triangle--the angle you're looking at and the 
right angle; these determine the third angle (since the three angles 
add to 180 degrees), and the triangle is determined up to a scale.  By 
this I mean that any two triangles with these same angles will be 
similar, so the ratios of corresponding sides will be the same.

If you try to apply the definitions to the right angle in the right 
triangle, you only know one angle of the triangle, so the ratios of 
sides are not fixed.

Thus, if you want to think about the trig functions of a right angle 
in terms of the triangle-based definitions, you need to picture a 
triangle with TWO right angles.  Such a triangle does not exist!  Two 
sides would be parallel, so they never meet to form the third vertex.

You can consider what happens when the angle gets closer and closer to 
a right angle but isn't quite a right angle.  We call this the LIMIT 
of the trig function as the angle approaches 90 degrees.

Let's say we're looking for the tangent of angle A; B is the right 
angle, and side AB is 1 unit long.  Then tan(A) = BC/AB = BC, since 
AB = 1.  As angle A gets closer and closer to 90 degrees, the vertex 
C gets farther and farther out, so BC increases without limit.  
Loosely, we can say that tan(A) is infinity, or infinitely large.  
It's better, though, to say that it is undefined.

Have you learned anything about defining trig functions of angles 
outside the range 0 to 90 degrees, using the unit circle?  If not, 
wait for that.  It will give you different perspectives on your 
question.  You'll see that there is no problem defining sin(90) and 
cos(90).  On the other hand, you'll see that the tangent of an angle 
very close to 90 degrees can be either a very large positive number 
or a very large negative number.  That's one reason why I say that it 
isn't a good idea to call tan(90) "infinity".

- Doctor Rick, The Math Forum 

Date: 09/22/2006 at 17:41:33
From: Alex
Subject: Thank you (What the opposite and adjacent sides of a 90
degree angle)

Thank you for answering my question.  No teacher actually stated
clearly that the tan, sin, and cosine of a right triangle, defined in
terms of opposite, adjacent, and hypotenuse, are only for the acute
Associated Topics:
High School Trigonometry

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