Associated Topics || Dr. Math Home || Search Dr. Math

### Rotating a Point around an Angle

Date: 05/16/97 at 14:09:52
From: ALEXANDER CHUPACHENKO
Subject: X1=Cos(angle)*X - Sin (angle)*Y

Hello,

Could you explain this formula to me?

X1 = Cos(angle)*X - Sin (angle)*Y
Y1 = Sin(angle)*X + Cos (angle)*Y

I know when to use it, but I would like to know how it works.

Thank you very much,
Alexander

Date: 05/20/97 at 10:06:57
From: Doctor Mitteldorf
Subject: Re: X1=Cos(angle)*X - Sin (angle)*Y

Dear Alexander,

The formulas:

x1 = x*cos(t) - y*sin(t)
y1 = y*cos(t) + x*sin(t)

are useful for rotating a point (x,y) in the XY plane by an angle t.

After the rotation, we have a new point (x1,y1) whose coordinates can
be calculated from the old point.

(In other problems, you think of the same thing as leaving the point
fixed and rotating the coordinate system through an angle -t.)

You can derive these formulas just using a little geometry, and the
definitions of sin and cos.  Just draw yourself a diagram with a point
in the first quadrant (x,y) and rotate it to a new point, draw all the
triangles - including the little one on the right.  I'll bet you can
get it yourself. Write back if you have any trouble.

You might also notice that these formulas are closely related to the
formulas for the sin and cos of the sum of two angles.  If you let

x = cos(p) and y = sin(p)

Then the formulas just say:

cos(t+p) = cos(t)cos(p) - sin(t)sin(p)
sin(t+p) = cos(t)sin(p) + sin(t)cos(p)

Please write again and let me know how you did with the geometric
derivation.

-Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

Associated Topics:
High School Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search