Mathematical Investigations IV
Name:
Mathematical Investigations IV
Trigonometry - Beyond the Right Triangles
Revisiting the Rotation Matrix
Remember the rotation matrices from MI-2? Let’s review.
If we take a point ( x,y ) and write it as a column matrix x
 
, we can use matrix operations to move this point around the plane. Actually, these matrix operations transform all points in the plane.
1.
Look at the matrix

1 0
 0 1


. Then for any
This matrix is called the “Identity”. Why? x
 
in the plane,

1 0
 0 1
 x

?
2.
In MI-2 we derived the matrix that rotates the plane 90
 counter-clockwise about the origin. This matrix is R
90


 a b c d
__ __
__ __


. If you don’t remember, we should be able to figure this out.
For example, if the point (1,0) is rotated 90
 counter-clockwise about the origin, it would rotate to the point ( ____ , ____). (1)
Similarly, if the point (0,1) is rotated 90
 counter-clockwise about the origin, it rotates to the point ( ____ , ____). (2)
So we want

 a b c d 0
 a c

  



from (1) and

 a b c d 1
 b d

  



from (2).
Then, R
90


 a b c d
__ __
__ __


If you want a little more practice, use the above technique to find matrix that rotates the plane 270
 counter-clockwise about the origin or reflect the plane over the line y = x .
Trig. 7.1
Rev. F13

Mathematical Investigations IV
Name:
3.
Now we’re ready to try to find a matrix by R

R

 a b c d rotates the point through an arbitrary angle


, such that multiplying a point

about the origin in the counterx
  clockwise direction. a.
To do this, let's consider the unit circle and right-triangle trigonometry. Label point P in terms of angle

. This point can be considered the result of rotating the point (1, 0) through an angle

about the origin in the counter-clockwise direction. So let’s use this to derive a matrix R

 

 a b c d


, by starting with the point
1
 
.

P
Take the point matrix R

1
 
 a b c d


and multiply it by the
:
( _____ , _____ )
 a b
 c d

1

( 1,0)
Set your result equal to the coordinates of point P you found above. This should give you two of the entries in matrix R

.
Trig. 7.2
Rev. F13

Mathematical Investigations IV
Name:
To determine the remaining entries in matrix R

, repeat the process with the point
0
 
.
Rotate this point through an angle

about the origin in the counter-clockwise direction to point Q. Label this point in terms of angle

: Q (______, ______)
(Hint: Look at congruent triangles.)
Q ( _____ , _____ )

(0,1)

P (cos , sin )
If you aren't sure how to label point Q in terms of angle

, consider this. Point Q is a rotation of point P in the counter-clockwise direction about the origin. What is the degree measure of this rotation? ___________. If you are thinking correctly, you should be able to use a matrix from problem 2 to rotate point P onto point Q via matrix multiplication to determine the coordinates of point Q in terms of angle

. Perform the appropriate operation here:
Next, Take the point
0
 
and multiply it by the matrix R


 a b c d

0
1

 a c b d


:
Finally, set your result equal to the coordinates of point Q you found above. This should give you the remaining two of the entries in matrix R

.
So,
R



 a b c d


  



Trig. 7.3
Rev. F13

Mathematical Investigations IV
Name:
4.
Find R
30 and simplify the result.
5.
Use your result from problem 4 to rotate the point




4

4 3


 about the origin.
30 counter-clockwise
6.
Let’s try to rotate the line y

3 x

4 about the origin through an angle of 90 .
It might make it easier to represent this line using parametric equations: x
 t y 3 t 4
. a.
Since every point on the line has the form x
   t
3 t

4


, we need only multiply this
“point” by the matrix
R
90
. Find the result. b.
Next, find the relationship between the two coordinates of the new point to get the equation of the rotated line. c.
Then graph both lines on the axes to the right:
7.
Now rotate the line y

3 x

4 about the origin through an angle of 30 , and write the equation of the rotated line.
Trig. 7.4
Rev. F13