An animation of **the
formula for a plane rotation, (ax-by, bx+ay)**:

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If $(1,0)$ is rotated about the
origin to $(a,b)$,
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then
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$(x,y)$ is rotated to $(ax-by,bx+ay)$
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(Place your mouse over the steps
in each derivation to see the justifications)

Both of **the trigonometric
addition formulas**

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$\cos(s+t) = \cos s \cos t - \sin s \sin t$ and $\sin(s+t) = \cos s
\sin t + \sin s \cos t$
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follow immediately

by letting
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$(a,b) = ( \cos s , \sin s )\hbox{ and }(x,y)=( \cos t , \sin
t ).$
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See how the graphs of the **cosine**
and **sine** functions,
and **both simultaneously,** are

related to the horizontal and vertical components of uniform rotation
at these links.

**The Pythagorean relationship**,
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$a^2+b^2=1\hbox{ or } \cos^2 s + \sin^2 s =1$
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also follows by letting
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$(x,y)=(a,-b)\hbox{ since }(a,-b)$
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is rotated to
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$(1,0)$
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.

Multiplication of **complex numbers,
**
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$(a+bi)(x+iy)=(ax-by)+(bx+ay)i,$
with
$i^2=-1 ,$ is instantly seen to represent the rotation formula,
plus a scaling that allows
$(1,0)$
to go anywhere.
More {\bf links} to other connections under construction:
The derivation is based upon two congruent triangles and
two similar triangles from Euclidean geometry.
The derivation uses linear superposition of basic problems involving
1's and 0's to solve more general problems, the same strategy that is used
in Lagrange interpolation and
term-by-term differentiation
and term-by-term integration.
Euler's Formula: Uniform circular
motion is characterized by a relationship between velocity and position:
The velocity vector is given by rotating the position vector one quarter
of a circle (counterclockwise by convention.) This is either written
${dx\over dt}=-y {dy \over dt}=x\hbox{ or }{dz \over dt}=iz.$
The solution of the
latter starting from
$z=1\hbox{ at }t=0\hbox{ is }e^{it}= \cos t + i \sin t.$
The
differential equations framework is very convenient for making the
rotation formula and its consequences rigorous. From this point of view,
the relations
$\exp'=\exp, \cos'=-\sin,\hbox{ and }\sin'=\cos$
are the defining
properties of these functions, not formulas to be derived.
Invariance and geometric interpretation of the dot and cross products:
If two vectors are simultaneously rotated about the origin, the rotation
formula and Pythagorean relationship show that their dot
product remains unchanged. By rotating both to $(r_1,0)$ and
$( r_2 \cos s , r_2 \sin s )$ we see that the dot product represents $r_1
r_2 \cos s$ , the product of the magnitudes times the cosine of the angle
between the vectors. This generalizes to three and more dimensions, and
to the {\bf cross product} as well.
Orthogonalization, QR decomposition:
The geometric interpretation of the dot product above was based upon
the constructively demonstrable fact that two vectors in the plane may
be simultaneously
rotated so that all but the first component of the first vector is
zero. This method has a powerful generalization. In three dimensions, we
may use plane rotations to simultaneously rotate three vectors so that
all but the first component of the first vector is zero $(r_1,0,0)$, and
all but the first two components of the second are zero $(r_2 \cos s ,
r_2 \sin s , 0)$. In $n$ dimensions, we may construct a sequence of plane
"Givens rotations" which when applied simultaneously to $k \leq
n$ vectors, makes all but the first $j$ components of the $j$th vector
equal to zero. This may be also be viewed as orthogonalizing a set of vectors
with triangular coefficients, the same effect as the Gram-Schmidt process.
The Cauchy-Riemann equations:
The form of the Cauchy-Riemann equations of complex analysis
is the same as that of the coefficients of $x$ and $y$ in the rotation
formula: ${\partial u \over \partial x}={\partial v \over \partial y}$
says that the coefficient of $x$ in the first component equals the coefficient
of $y$ in the second component (the two "$a$"s) and ${\partial u \over
\partial y}=-{\partial v \over \partial x}$ says the the coefficient of
$y$ in the first component is "-" the coefficient of $x$ in the second
component (the $b$ and the -$b$).
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