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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/
```
Associated Topics:
High School Trigonometry

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