Let's break down the problem step-by-step.

1. Initial Setup

• The wheel has a radius of 1 foot.
• The center is at the origin: (0, 0).
• There is a paint spot initially at the coordinate (1, 0), which is on the rim of the circle.

2. Understanding the Rotation

When a point on the circle is rotated by an angle $\theta$ (in degrees) in the counterclockwise direction, the new coordinates can be calculated using basic rotation formulas.

3. Rotation Formula

For a point $(x, y)$ on a circle centered at the origin, if rotated by an angle $\theta$ counterclockwise, the new coordinates $(x', y')$ are given by:

$x' = x \cos \theta - y \sin \theta$ $y' = x \sin \theta + y \cos \theta$

4. Applying the Formula for $\theta = 27^\circ$

Initially, the paint spot is at $(x, y) = (1, 0)$. Substituting $x = 1$ and $y = 0$:

$x' = 1 \cdot \cos 27^\circ - 0 \cdot \sin 27^\circ = \cos 27^\circ$

$y' = 1 \cdot \sin 27^\circ + 0 \cdot \cos 27^\circ = \sin 27^\circ$

Therefore, after a rotation of 27 degrees, the coordinates of the paint spot are:

$(\cos 27^\circ, \sin 27^\circ)$

5. General Formula for Any $\theta$

If the wheel is spun by an angle $\theta$ in degrees, the coordinates of the paint spot become:

$(\cos \theta, \sin \theta)$

This formula is derived from the rotation transformation for any given angle $\theta$.

6. Numerical Values for 27 Degrees

Calculating the cosine and sine values for 27 degrees:

• $\cos 27^\circ \approx 0.8910$
• $\sin 27^\circ \approx 0.4540$

Hence, the exact coordinates after a 27-degree rotation are approximately:

$(0.8910, 0.4540)$

Summary

• After a 27-degree rotation, the coordinates of the paint spot are $(0.8910, 0.4540)$.
• For any general angle $\theta$, the coordinates are $(\cos \theta, \sin \theta)$.

Let me know if you'd like to explore this further or see more details!

Related Questions:

1. What would the coordinates be if the wheel was rotated by 90 degrees?
2. How does the formula change if the wheel is rotated in a clockwise direction?
3. What happens if the paint spot was initially at $(-1, 0)$?
4. Can you derive the formula if the center of the circle is at $(h, k)$ instead of the origin?
5. What are the coordinates of the paint spot if rotated by $180^\circ$?

Tip:

When dealing with rotations, always remember that counterclockwise rotation corresponds to positive angles and clockwise rotation to negative angles in trigonometric terms.