What is the equation of the curve?

What is the equation of the curve which is sinusoidal about the line y = x?

• In simpler words, the curve is rotation of y = sin x rotated about the origin by 45 degrees. In general, how do we find equations of curves which are rotations of standard curves?

• Answer:

  Let the coordinate system be rotated anticlockwise so the the line given by y=xy=xy=x lies on XXX axis (with YYY perpendicular to it). So, we now have the original coordinate system with xxx and yyy axis and the rotated coordinate system XXX and YYY. We have (θ=45∘θ=45∘\theta=45^\circ), X=xcosθ+ysinθ=x2√+y2√X=xcos⁡θ+ysin⁡θ=x2+y2X=x\cos\theta+y\sin\theta=\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}} and Y=−xsinθ+ycosθ=−x2√+y2√Y=−xsin⁡θ+ycos⁡θ=−x2+y2Y=-x\sin\theta+y\cos\theta=-\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}. The inverse relation would be: x=Xcosθ−Ysinθ=X2√−Y2√x=Xcos⁡θ−Ysin⁡θ=X2−Y2x=X\cos\theta-Y\sin\theta=\frac{X}{\sqrt{2}}-\frac{Y}{\sqrt{2}} y=Xsinθ+Ycosθ=X2√+Y2√y=Xsin⁡θ+Ycos⁡θ=X2+Y2y=X\sin\theta+Y\cos\theta=\frac{X}{\sqrt{2}}+\frac{Y}{\sqrt{2}}. A general equation of the curve is (AAA is a constant): Y=sin(X−A)Y=sin⁡(X−A)Y=\sin(X-A) ⇒−x2√+y2√=sin(x2√+y2√−A)⇒−x2+y2=sin⁡(x2+y2−A)\Rightarrow -\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}=\sin\left (\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}-A \right )