Tangent

Date: 02/14/2003 at 10:10:10
From: Laura
Subject: Trigonometry

Is the trigonometric ratio tangent related to the tangent of a circle?

Date: 02/14/2003 at 10:21:20
From: Doctor Jerry
Subject: Re: Trigonometry

Hi Laura,

Yes. Draw a coordinate axis - just as when you sketch an (x,y)-plane.  
Draw a unit circle, centered at the origin. Draw a tangent to this 
circle, upward from B=(1,0). Draw a line L from (0,0) to this tangent 
line - do this so that L is in the first quadrant. Let A be the point 
where L meets the tangent line. The line  L makes an angle theta with 
the x-axis. Note that tan(theta) = AB/1 = AB.

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

Date: 02/17/2003 at 10:44:27
From: Laura
Subject: Trigonometry

Having read about the relation between the tangent of a circle and the 
tan ratio used in trigonometry, I wondered if the sine and cosine 
ratios were also linked to circles? If so, what about cotangent, 
secant and cosecant?

Date: 02/17/2003 at 12:10:04
From: Doctor Jerry
Subject: Re: Trigonometry

Hi  Laura,

Yes, one can show that sine and cosine have natural interpretations 
(see below); maybe one could give interpretations for cotangent, 
secant, and cosecant (each of this is the reciprocal of sine, cosine, 
or tangent), but I believe the interpretations would be somewhat 
artificial and not terribly useful.

For sine and cosine, draw a coordinate axis, just as when you sketch 
an (x,y)-plane. Draw a unit circle, centered at the origin. Draw a 
line L from (0,0) to the unit circle. It can be in any quadrant. Let 
theta be the angle needed to rotate the positive x-axis 
counterclockwise until it conincides with L. Let A=(a,b) be the point 
where L intersects the unit circle.

It is quite easy to see that a = cos(theta) and b = sin(theta).

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

Date: 02/17/2003 at 13:58:34
From: Laura
Subject: Thank you (Trigonometry)

Thank you very much for your help in answering my questions. Maths is 
my favourite subject and I quite often drive my teacher up the wall 
with all my "but why?"'s. I have been able to follow your instructions 
and prove how it works for myself. I will show my teacher this week. 
Thank you.