Defining the Six Trigonometric Functions on the Unit Circle

Date: 09/11/2003 at 14:02:08
From: Cathy
Subject: the six trigonometric functions 

I can find sin, cos, and tan on the unit circle, but I don't know how
to find csc, cot, and sec.

Date: 09/11/2003 at 15:31:53
From: Doctor George
Subject: Re: the six trigonometric functions 

Hi Cathy,

Thanks for writing to Doctor Math.

Let L be the line through the origin at angle A relative to the x 
axis.

For the cosine, let P be the intersection of L with the unit circle. 
The cosine is the distance from (0,0) to the x coordinate of P, and 
the sine is the distance from (0,0) to the y coordinate of P.

For the tangent, let T be the line tangent to the unit circle at P. 
The segment of T from P to the x axis length tangent. The segment of 
T from P to the y axis has length cotangent.

There is another way to draw these two. For the tangent, let P1 be 
the intersection of L with the line x = 1. The segment from P1 to 
(1,0) has length tangent. For the cotangent, let P2 be the 
intersection of L with the line y = 1. The segment from P2 to (0,1) 
has length cotangent.

For the secant, let Q be the intersection of L with the line
x = 1. The segment from (0,0) to Q has length secant.

For the cosecant, let R be the intersection of L with the line
y = 1. The segment from (0,0) to R has length cosecant.

If you apply the Pythagorean theorem to the right triangles we 
created you will see other trigonometric identities.

Does that make sense? Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/