Thread: the rotation of a sinusoid curve

1. the rotation of a sinusoid curve

Hello,
I have a sinusoid curve in an euclidian orthonormal plane [OX,OY].
the equation of the curve is y=a sin x.

Now, I want to get the equation of the curve when I rotate it by an $\alpha$ angle with respect to the (ox) axis like it is shown in the figure submitted.

To do that,I'll define tehe vectorial euclidian rotation :in an oriented vectorial euclidian plan, a vectorial rotation is simply defined by its angle $\alpha$. Its matrix in an orthonormal direct basis is :
$
\begin{pmatrix}
\cos\alpha & -\sin \alpha\\
\sin \alpha & \cos\alpha
\end{pmatrix}$

In another way, a vector $\vec U$ (x,y) has as image the vector $\vec V$ (x',y') that can be calculated like that :

$
\begin{pmatrix}
x'\\y'\end{pmatrix} =\begin{pmatrix}\cos\alpha&-\sin\alpha \\ \sin\alpha&\cos\alpha\end{pmat rix}\begin{pmatrix}x\\y\end{pm atrix},
$

then we have :
$x' = x \cos \alpha - y \sin \alpha\,$
and
$y' = x \sin \alpha + y \cos \alpha\,$

as we have

$
y=a \sin x$
, then we have

$y' = x \sin \alpha + a \sin x \cos \alpha\,$

the problem is that I must write $y'$ in function of $x'$.

Thank you for reading me. I'll be vey glad if you can help me

2. Were you given restrictions on $\displaystyle \alpha$? Because for $\displaystyle \alpha > 45^o$, y' cannot be expressed as a function of x' (since it fails the vertical line test).

3. Hello Snowtea,

thank you for answering but I didn't understand why I must give restrictions to ?
I tried to solve the problem like that but It seems to me complicated:

As we have y=a sin x.

and if we rotate the vector v(x',y') in an angle -, we have then u(x,y) and the matrix of rotation is:

$\displaystyle$\begin{pmatrix}
x\\y\end{pmatrix}=\begin{pmatrix}-\cos\alpha&-\sin\alpha \\ \sin\alpha&-\cos\alpha\end{pmatrix}\begin{pmatrix}x'\\y'\end{p matrix},

then we have

$\displaystyle$x = -x' \cos \alpha - y' \sin \alpha\,
and
$\displaystyle$y = x' \sin \alpha - y' \cos \alpha\,$$then we replace x and y in the equation y=a sin x by their expressions in funtion of x' and y'. we then have: \displaystyle$$ x' \sin \alpha - y' \cos \alpha\ = a sin(-x' \cos \alpha - y' \sin \alpha)$The problm now is how to write the equation in a simple form y'=f(x'). Could you help me. Thank you 4. You can only write y' = f(x') when for every x', you have a unique y' that satisfies the equation. If$\displaystyle \alpha > \pi/4$, just consider the simple case when$\displaystyle \alpha = \pi/2$. Then for$\displaystyle x'=0$,$\displaystyle y'\in\{0, \pi, -\pi, ...\}\$ are all solutions, so y' cannot be written as a function of x'.
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