A wheel of radius one foot is placed so that its center is at the origin, and a paint spot

on the rim is at (1, 0). The wheel is spun 27 degrees in a counterclockwise direction. What are

the coordinates of the paint spot? What if the wheel is spun θ degrees instead?

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.

A wheel of radius one foot is placed so that its center is at the origin, and a paint spot on the rim is at (1, 0). The wheel is spun 27 degrees in a counterclockwise direction. What are the coordinates of the paint spot? What if the wheel is spun θ degrees instead?

Learn how to find the new coordinates of a point on a wheel's rim after a counterclockwise rotation. Starting with an initial position of (1, 0), we explore the formulas for calculating the new position after rotating by 27 degrees or any general angle θ. Suitable for high school students in Grades 10-12.

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