### Rotation Matrices

#### Part 1: Trigonometric Background

- Let
**(x,y)** denote the terminal
point of the vector **v** in the plane. Suppose we rotate this vector
through an angle
and let **(x**_{r},
y_{r}) denote the new terminal point **v**_{r} after
the rotation. Our first objective is to calculate the new coordinates
in terms of the old ones. If **b = |****v**|, then **x = b cos()**
and **y = b sin()**. Similar formulas
give us **x**_{r} and **y**_{r} in terms of the angle ** +
**. Enter the formulas for **x**_{r}
and **y**_{r}, and use the expand function to rewrite these formulas
in terms of the angles and .
Then identify occurrences of **x** and **y** in these expressions, and rewrite
the formulas to give **x**_{r} and **y**_{r} in terms of **x**, **y**,
and .
- Rewrite the transformation
(rotation through an angle ) as
a matrix-vector equation,
**v**_{r} = **Av**. That is, give
the matrix **A**.

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