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Definition of a Tangent

Date: 10/27/95 at 23:46:22
From: Anonymous
Subject: What is a tangent of an angle?

I am a senior and I have already taken Geometry, but for some reason I 
can't seem to remember what the tangent of an angle is.
I have asked my parents and looked in a few encylopedias and 
dictionaries, but I haven't been able to find out which one it is.
I wouldn't be asking, but I have an interview for a job tomorrow and the 
boss asked me what the tangent of an angle is.

Q: What is the tangent of an angle?

Thanks for your time I am sorry to have to bother you.

Date: 10/28/95 at 11:35:6
From: Doctor Ken
Subject: Re: What is a tangent of an angle?


I hope this response reaches you in time.  The tangent of an angle can 
be thought of in a couple of different ways, one of which is easy to 
think about, the other of which is a little more correct and a little 
more general. They're essentially the same thing, though.

1) If you're dealing with right-triangle trigonometry, then the tangent 
of an angle (not the right angle) is the ratio of the opposite side to 
the adjacent side.  For instance, if you have a right triangle whose 3 
sides have length 1, Sqrt{3}, and 2, the tangent of the angle opposite 
the side of length 1 is 1/Sqrt{3}.  Look at the picture and justify that 
to yourself.  By the way, the angle in question is 30 degrees.

        | \
        |  \ 2
Sqrt{3} |   \ 
        |    \

2) The somewhat more rigorous way.  Tangent is actually defined in terms
of Sine and Cosine.  It's defined as Sine/Cosine.  So then you have to 
ask "what are Sine and Cosine defined as?"  Well, they're defined in 
terms of a circle of radius 1.  Do the following: draw a circle of 
radius 1 that's centered at the origin of a coordinate system.

Then draw a ray that starts at the origin and makes an angle b with the 
x axis (you go around counterclockwise from the x axis until you hit the 
ray).  Cosine and Sine are the coordinates of where that point 
intersects the circle.  Cosine is the x coordinate, and Sine is the y 

So then once you have Sine and Cosine, you can divide them to get 

-Doctor Ken,  The Geometry Forum

Associated Topics:
High School Definitions
High School Trigonometry

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