Rotation Matrix: Finding Two Expressions & Verifying Equivalence

[yoyo]

Homework Statement

A vector x in R^2 is rotate twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify that these two expressions are equivalent

Homework Equations

rotation matrix R=[cos, -sin; sin, cos]

The Attempt at a Solution

I can only think of one expression:

(R)(R)(x).

Could (R^2)(x) be the other one? How would i prove that this is equivalent?

 

[ZioX]

How about just rotating once by 2theta?

 

[yoyo]

follow up question: a vector x in R^2 is rotated n times through an angle theta. Find two expressions for the matrix representing this rotation. what identity is implied.

if what Ziox said is true then it should be [cos (theta)n, -sin(theta)n; sin (theta)n, cos(theta)n]

but i don't see what identity this implies?

 

[cristo]

follow up question: a vector x in R^2 is rotated n times through an angle theta. Find two expressions for the matrix representing this rotation. what identity is implied.

if what Ziox said is true then it should be [cos (theta)n, -sin(theta)n; sin (theta)n, cos(theta)n]

but i don't see what identity this implies?

You should probably find a second expression for the matrix first.

 

[HallsofIvy]

First, what is
$$\left(\begin{array}{cc}cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta)\end{array}\right)^2$$?

Second, can you use trig identities to write that in terms of $cos(2\theta)$ and $sin(2\theta)$?