The image below suggests that there is a small, positive value of $x$ such that

1. Use the small angle approximation to estimate the coordinates of $P$.

2. In the image below, $y=\sqrt{x}$ has been translated to the right by $k$ units, and $y=\sin (2x)$ remains unchanged. The image suggests that there should be some value of $k$ such that the curves intersect at exactly one small value of $x.$ Approximate the value of $k.$

For part (a), because $2x$ is small, you can have

You can square both sides of the equation before rearranging and solving.

For part (b), the translated curve has equation $\sqrt{x-k}.$

Square both sides and use the discriminant.