Calculator: TI-83 Plus.
A matrix [B] = [[cos(A), -sin(A)][sin(A), cos(A)]] represents a counterclockwise rotation by angle A, around a center of coordinates on a plane. Remember that if A is negative, the direction is actually clockwise (opposite to counterclockwise).
Here is a program that rotates a 2-dimensional picture counterclockwise by angle A around the origin. It uses (calls) the program DRAW, which is in the unit Pictures. You need to prepare matrices [I] and [J] beforehand. [I] holds the coordinates of the points in your drawing, and [J] tells which points are connected. The program keeps all the pictures, overlaying the next one on the previous ones. (I named it ROTATETW (for "rotate in 2 dimensions"), but you may choose your own name!)
| Drawing Program | Explanation |
| PROGRAM:ROTATETW | |
| :PrgmDRAW | Draws picture in original orientation |
:Prompt A :While 1 :[[cos(A), -sin(A)][sin(A),cos(A)]]→[B] |
User enters angle A in degrees |
| :[B]*[I]->[I] | Rotate matrix [I] counterclockwise by angle A |
| :PrgmDRAW | Draw the new picture |
| :Pause | Press enter to get the next picture |
| :End |
How would you change the program if you wanted to erase the previous picture before drawing the next one?
The question is: Given a 2 by 2 matrix [B], does it represent a rotation? If so, how to find its angle A?
The answer to the first question.
If det([B]) ≠ 1, then [B] is not a rotation,
else if [B]*[B]T is the 2 by 2 identity matrix [[1,0][0,1]] then [B] is a rotation,
else [B] is not a rotation.
The answer to the second question.
If [B] is a rotation, compute, cos-1([B](1,1)), (it is always positive); then if [B](2,1) is negative then A = -cos-1([B](1,1)), else A = cos-1([B](1,1))
Task.
Students work in pairs using one calculator. One student prepares a matrix which either is or is not a rotation, and the other checks it (and finds A if appropriate).(Set: MODE to Float 2.)
Example.
First student:
[[cos(27),-sin(27)][sin(27),cos(27)]]→[B]
[[.89 -.45]
[.45 .89 ]]
CLEAR
Second student:
det([B])
1.00 So far it is okay.
[B]*[B]T
[[1 0]
[0 1]] It is a rotation.
cos-1([B](1,1))
27.00
[B](2,1)
.45 It is positive.
The angle is A = 27 degrees.
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