Question

rotate the point 137° to within 10°.

Explanation:

Step1: Determine the rotation direction

Since the problem does not specify clockwise or counter - clockwise, we assume counter - clockwise (standard in mathematics unless stated otherwise).

Step2: Calculate the equivalent angle within one full rotation ($360^{\circ}$)

We know that angles in standard position repeat every $360^{\circ}$. Let's find an equivalent angle $\theta$ of $137^{\circ}$ such that $\theta = 137^{\circ}+k\times360^{\circ},k\in\mathbb{Z}$. But if we consider the concept of "within $10^{\circ}$" in terms of coterminal angles (angles that have the same terminal side). Another way is to consider the reference angle. The reference angle $\alpha$ of an angle $\theta$ in standard position is given by $\alpha=\theta$ if $0\leq\theta < 90^{\circ}$, $\alpha = 180^{\circ}-\theta$ if $90^{\circ}<\theta<180^{\circ}$. For $\theta = 137^{\circ}$, the reference angle $\alpha=180 - 137=43^{\circ}$. If we consider the closest angle to a multiple of $90^{\circ}$ within $10^{\circ}$.
Let's check the distance from $137^{\circ}$ to $135^{\circ}$ (which is $\frac{3\pi}{4}$ radians or $135^{\circ}$) and $140^{\circ}$.
The distance from $137^{\circ}$ to $135^{\circ}$ is $|137 - 135|=2^{\circ}$.
The distance from $137^{\circ}$ to $140^{\circ}$ is $|137 - 140| = 3^{\circ}$.

Answer:

The point rotated $137^{\circ}$ is closest (within $10^{\circ}$) to a rotation of $135^{\circ}$ (or $\frac{3\pi}{4}$ radians).