Rotation
The Trigonometry –The general rotation matrix uses the trigonometric functions, sine and cosine of the angle
of rotation. Students are generally introduced to right triangle trigonometry in the third quarter of geometry. This
means they understand the meaning of the trigonometric ratios sine, cosine, and tangent for acute angles in right
triangles. In later courses students are introduced to the unit circle which enables them to expand the domain of the
trigonometric functions to all real numbers. This is a good point to preview the upcoming material. Tell students
that they will shortly learn a method for finding sin 90◦and cos 120◦; let them know that it is a brilliant method used
to expand these extremely useful definitions. For now though, they will have to trust the number given to them by
their calculator when using the general rotation matrix. It is not practical to give them the full explanation now, but
letting them know that there is an explanation, and that they will learn it soon, will avoid confusion.
Degrees or Radians –If students have followed standard high school mathematical curriculum, they have no idea
what a radian is, but somehow calculators frequently end up in radian mode. Students are not familiar enough with
trigonometric functions to realize that they are not getting the correct ratio, and continue with their calculations
resulting in incorrect answers. Show the students how to check if their calculators are in radian or degree mode
and how to change the mode. This is also a good opportunity to start students thinking about different ways to
measure angles. Let them know that radians are another unit used to measure rotation. Similar to how both inches
and centimeters both measure length. Making students aware of radians now, will help them avoid errors and ready
their minds to learn about radians in the future.
Additional Exercises:
- Graph 4 ABCwithA( 2 , 1 ),B( 5 , 3 ), andC( 4 , 4 ).
- Put the vertices of the triangle in a 3×2 matrix and use matrix multiplication to rotate the triangle 45 degrees.
Graph the image of 4 ABCon the same set of axis using prime notation. - Use the matrix multiplication to rotate 4 A[U+0080][U+0099]B[U+0080][U+0099]C[U+0080][U+0099]60 degrees.
To do this, take the matrix produced in #2 and multiply it by the rotation matrix for 60 degrees. Graph the resulting
triangle on the same set of axis as 4 A[U+0080][U+0099][U+0080][U+0099]B[U+0080][U+0099][U+0080][U+0099]C[U+0080][U+0099][U+0080][U+0099]. - What single matrix could have rotated 4 ABCto 4 A[U+0080][U+0099][U+0080][U+0099]B[U+0080][U+0099][U+0080][U+0099]C[U+0080][U+0099][U+0080][U+0099]
in one step? How does this matrix compare to the 45 degree and 60 degree rotation matrix?
Answers:
2.
2 1
5 3
4 4
[
. 707. 707
−. 707. 707
]
=
.707 2. 12
1 .41 5. 66
0 5. 66
3.
.707 2. 12
1 .41 5. 66
0 5. 66
[
. 5. 866
−. 866. 5
]
=
− 1 .48 1. 67
− 4 .20 4. 05
− 4 .90 2. 83
4.
[
− 0. 26 1. 00
− 1. 00 − 0. 26
]
=
[
. 707. 707
−. 707. 707
][
. 5. 866
−. 866. 5
]
Composition
Use the Image –When first working with compositions student often try to apply both operations to the original
figure. Emphasis that a composition is a two-step process 83s. The second step of which is performed on the result
2.12. Transformations