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Rotating a Point around an Angle

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


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,

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 

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Trigonometry

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