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### 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?

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

```
Associated Topics:
High School Trigonometry

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