Graph of $\sin(x)$ along the line $y=x$

Question

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.

Answer 1

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!

Answer 2

The curve is given implicitly by the equation $$ \frac{-x+y}{\sqrt2}=\sin\frac{x+y}{\sqrt2} . $$ You won't be able to solve for $y$ as an explicit function of $x$.

(Wolfram Alpha plot.)