Graphs of Sine and Cosine Functions

Date: 02/28/2003 at 17:22:35
From: Nati
Subject: Relation between the sine and the cosine function

How is the graph of y = cos(x) linked to the graph of y = sin(x)?

The Pythagoran Identity, meaning sin^2 x + cos^2 x = 1, is what I 
found out about the relation concerning sine and cosine. But aren't 
there any general explanations other than the fact that the sine is 
just a displaced cosine curve along the x-axis ?

Date: 02/28/2003 at 17:31:21
From: Doctor Tom
Subject: Re: Relation between the sine and the cosine function

Hi Nati,

Basically you are right - the only difference between the sine and the 
cosine is that they are shifted versions of each other.

In a right triangle, you have the right angle and two other angles A 
and B. Since A and B together make 90 degrees, A is the complement of 
B and vice-versa.

The (co)sine of an angle is is the sine of the (co)mplement. In other 
words, the sine of A is the cosine of B and the sine of B is the 
cosine of A.

The same idea holds for the other trigonometric functions: the 
cotangent is the tangent of the complement and the cosecant is the 
secant of the complement.

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

Date: 02/28/2003 at 22:57:05
From: Doctor Dotty
Subject: Re: Relation between the sine and the cosine function

Hi Nati,

Thanks for the question!

Lets go back to basics. Triangles:
 
  |.
  |     .   c
 b|_          .
  |_|_ _ _ _ _x_(_ _.
           a

The following equations are true:

          b
  sin x = -
          c

          a
  cos x = -
          c

But, suppose we put on another triangle:

   _ _ _ _a_ _ _ _ _
  |.              |_|
  |     .           |
 b|_      c   .   y |b
  |_|_ _ _ _ _x_(_ .|
          a 

Where y is the angle between c and the right hand b.

  x + y = 90

So    x = 90 - y.

We know that:

          b
  sin x = -
          c

If you look at the diagram, this gives:

                 a
  sin (90 - x) = -
                 c

Which is cos x.

So:

   sin (90 - x) = cos x

and therefore

   cos (90 - x) = sin x

This is why the graph of cos x is the graph of sin x displaced by 90 
degrees.

Note that if you are working in radians, 90 degrees is (Pi / 2).

Does that all make sense? 

If I can help any more with this problem or any other, please write 
back.

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