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 http://mathforum.org/dr.math/
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 angles.
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