I want the equation of $\sin(x)$ which has the line $y=x$ as its axis. Basically, I want the $\frac\pi4$ rotation of the curve $y=\sin(x)$.
I already attempted differentiating the curve and adding $\frac\pi4$ by using the angle between two lines formula, i.e., $\displaystyle \tan\theta=\frac{m_1-m_2}{1-m_1m_2}$. And consequently integrating the result.
Any help would be appreciated.
Best Answer
Hint:
Consider the parametrization of your curve:
$$\mathbf{s}(t) = (t, \sin(t)), \quad t \in \mathbb{R},$$
then, the transformation given by:
$$ \mathbf{p}(t;\theta) = M \, \mathbf{s}^T(t), \quad M = \left(\begin{array}{cc} \sin{\theta} & \cos{\theta} \\ -\cos{\theta} & \sin\theta \end{array}\right),$$
applies a clockwise rotation of angle $\theta$ to $\mathbf{s}(t)$.
Cheers!