Formulas for Sine and Cosine

Date: 1/24/96 at 21:23:51
Subject: What are the definitions of sin, cos, and tan in terms of theta?
From: Michael Allman

Dr. Math,

I have been searching for an answer for several days and 
am really stuck.  Even my precalc. teacher doesn't know (by the 
way - I am a high school junior.)  What are the definitions of sin, 
cos, and tan in terms of theta? So far, I have come upon a definition 
that looks something like:

   sin x = x + x^3/3! - x^5/5! + x^7/7! . . .

but I do not know how to handle this series.  Also, because of the 
ambiguous use of the variable x, I do not even know if this series is 
the answer I'm looking for.  So, is there a nice neat answer?

I also have a question about the notation of the trig. functions.  
If sin is a function and f(x) describes a function of x (e.g. f(x) = 
2x), why is sin theta defined by y and r in my textbook?  It seems to me 
that sin theta should be defined by theta and that sin (y,r) should be 
defined by y and r.

In a nutshell:

   1.  What is the definition of sin, cos, and tan in terms of the 
angle (not in terms of x, y, or r?)

   2.  Why is the function sin theta defined by y and r, and not by 
theta?

Thank you for your help!

Michael Allman

Date: 1/27/96 at 17:3:42
From: Doctor Ken
Subject: Re: What are the definitions of sin, cos, and tan in terms...

Hello!

The way Sine and Cosine are defined is usually in terms of the unit 
circle.  What you do is draw a circle of radius 1, whose center is at 
the point (0,0).  Then you draw a ray coming out from the origin that 
makes an angle of theta with the x-axis.  Note that the way you measure 
this angle is by starting at the x-axis, and travelling COUNTER-
CLOCKWISE until you hit the ray in question.  Thus this is about 60 
degrees:
    
     /
    /
   /
  /
 / theta
/_________________> x-axis

And this is about -60 degrees:

__________________> x-axis
\ theta
 \
  \
   \
    \
     \

This ray will intersect with the unit circle at a point (x,y).  
Cos(theta) is defined as x, the first coordinate of this intersection 
point, and Sin(theta) is defined as y, the second coordinate.

Then Tangent is defined as Sin/Cos, and so on.

>2.  Why is the function sin theta defined by y and r, and not by theta?

Well, I'm not really sure what you mean by y and r.  What are your y and 
r?

-Doctor Ken,  The Geometry Forum

Date: 1/29/96 at 15:1:2
Subject: Re: What are the definitions of sin, cos, and tan in terms of 
From: Michael Allman

Dr. Math,

You misunderstood my two questions.  The first one asked for the
definitions of the trigonometric functions in terms of theta, theta 
being the angle in question.  So if I needed to know the value of sin 
54.2 degrees and I did not have a calculator, how would I calculate 
this?  When you wrote "Cos(theta) is defined as x, the first coordinate 
of this intersection point, and  Sin(theta) is defined as y,...", you 
defined cos and sin in terms of the coordinates x and y.

As for my second question, y stands for the y coordinate of a 
point in a plane, and r stands for the length of the segment between 
that same point and the origin of the coordinate system.  See below.

         y-axis (x,y)
         |        /
         |       /
         |   r  /
         |     /
         |    /
         |   /
         |  /
         | /
    _____|/______________________ x-axis
  origin |
         |
         |

So, since sin theta is a function of theta, why is it defined by 
(y/r) and not by theta itself?  This seems to be an incongruity in 
function notation.

	I hope this is clearer.  Thank you.

Michael Allman

Date: 2/6/96 at 15:13:36
From: Doctor Ken
Subject: Re: What are the definitions of sin, cos, and tan in terms of 
theta? et al.

Hello!

Thanks, that is clearer.  First I should tell you that I think you don't 
actually mean "definition."  There is only one definition of Sine and 
Cosine of an angle, and that's what I gave you.  It sounds like what 
you're looking for is more of a "formula" for the Trig functions in 
terms of the angle, right?  In that case, the first few terms of the 
series you mentioned will work fine.

So for sin(x), a pretty good approximation will be
x - x^3/6 + x^5/120, and for cos(x) we have
1 - x^2/2 + x^4/24 - x^6/720.

If you don't need to be so accurate, you can drop the last term on each
of those series.

Another thing is that if you're going to try to take the Sine or Cosine 
of a theta that's not very close to 0, you should do some conversion 
first to improve your accuracy (these formulae are most accurate around 
0).  So if you were going to take the Cosine of 17Pi + 2.8, you'd say, 
"hey, that's about the same thing as 18Pi - .3415, which will have the 
same Sine as if theta were -.3415"; this gives you a MUCH better 
estimate.  In general, you want to get your angle between -Pi/2 and 
Pi/2.

Another thing to keep in mind: all you'll ever be able to find is an 
approximation like this.  For most angles (all except the special ones 
like Pi/3 and stuff) there is no neat, closed form expression for its 
Sine in terms of the original angle measure.

As for your second question, rest assured that Sine is well-defined.  
Each value of theta gives a unique point of intersection with the unit 
circle.  Sure, it's defined in terms of y/r (in your notation), but the 
y and the r are determined by theta.  It seems to me that you're 
confusing the notion of "definition" and "formula"; if you were 
expecting the definition of Sine to be a bunch of square roots and 
plusses and stuff, all in terms of theta, then that would explain your 
confusion.  But a definition can be whatever it wants to be: in terms of 
theta, or y, or r, or the color of the sky.

Hope this helps.

-Doctor Ken,  The Math Forum