## Rotations in the plane

What are the
coordinates of the point *(x', y')* we get
when we rotate *(x, y)* through an angle *a*?
The answer is
*
x' = x*cos(a) - y*sin(a)
*

y' = x*sin(a) + y*cos(a)

as the following figures illustrate.

In the first figure, the blue triangle
has vertex angle
*a*. If *r* is the radius of the circle,
then the radial side of the blue triangle is
*r cos(a)* and its circumferential side is *r sin(a)*.

Therefore in the following figure, the red, pink
and magenta triangles are all similar. The pink one is obtained from
the red one by a scaling factor of
*cos(a)*.
The magenta one is obtained from
the pink one by combining
(1) a rotation through a right
angle around one of its vertices and
(2) a suitable scale change.
The ratio of the size of the magenta one to the red one is *sin(a)*.

*Click and drag on the nodes to change (x, y) and a.*

If we choose *(x, y)* to be
of length *1*, say

*
x = cos(b)
*

y = sin(b)

then this gives us
the formula for the cosine and sine of
*a + b*.

*
cos(a + b) = cos(a) cos(b) - sin(a) sin(b)
*

sin(a + b) = sin(a) cos(b) + cos(a) sin(b)

*
*

*
*