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The vertices V and W of the arm are on a circle with diameter half the value (8 units) of the outer circle (16 units) and the line segments [SV] and [SW] both form an angle of 5° to ST2. What is the vertical distance between S and V?

To solve this, we can break it down step by step:

### Information provided:
- The outer circle has a diameter of 16 units, so its radius is \( R = 8 \) units.
- The vertices \( V \) and \( W \) are on a circle with a diameter of 8 units, meaning the radius of this inner circle is \( r = 4 \) units.
- The line segments \( [SV] \) and \( [SW] \) form an angle of \( 5^\circ \) to \( ST_2 \), which helps define the angular relationship between these points.
  
We are asked to find the **vertical distance** between \( S \) and \( V \), which implies finding the y-coordinate of \( V \) relative to \( S \).

### Step-by-Step Process:

1. **Position of \( S \) and \( T_1 \):**
   - Since \( T_1 \) is directly above \( S \), the vertical distance \( ST_1 \) is 8 units (radius of the outer circle).
   - So \( S = (0, 0) \) and \( T_1 = (0, 8) \).

2. **Position of \( T_2 \):**
   - Since \( T_2 \) is on the circle, its coordinates would be defined by some angle \( \theta \). Let's say it lies at a 0-degree angle from the horizontal (x-axis), meaning \( T_2 = (8, 0) \).

3. **Vertices \( V \) and \( W \):**
   - \( V \) and \( W \) are both on the inner circle, and the angle between \( [SV] \) and \( ST_2 \) is \( 5^\circ \).
   - Using this angle, we can place \( V \) using polar coordinates with respect to \( S \), since \( SV = 4 \) units (radius of the inner circle).
   
4. **Finding \( V \)'s Coordinates:**
   The polar coordinates of \( V \) relative to \( S \) are given by:
   \[
   V = (r \cdot \cos(5^\circ), r \cdot \sin(5^\circ))
   \]
   where \( r = 4 \) units, and \( 5^\circ \) is the angle from \( ST_2 \).

   Substituting the values:
   \[
   V_x = 4 \cdot \cos(5^\circ) \approx 4 \cdot 0.9962 = 3.9848 \text{ units}
   \]
   \[
   V_y = 4 \cdot \sin(5^\circ) \approx 4 \cdot 0.0872 = 0.3488 \text{ units}
   \]
   
5. **Vertical Distance:**
   Since \( S \) is at \( (0, 0) \) and \( V_y \) is the y-coordinate of \( V \), the **vertical distance** between \( S \) and \( V \) is simply the value of \( V_y \):
   \[
   \boxed{0.3488 \text{ units}}
   \]

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the horizontal distance between \( S \) and \( V \)?
2. How can we calculate the distance between \( S \) and \( T_2 \) using the coordinates?
3. What is the total distance between \( V \) and \( T_2 \)?
4. How can we compute the angle between \( ST_1 \) and \( ST_2 \)?
5. How would the solution change if \( V \) were located at a different angle than \( 5^\circ \)?

**Tip:** Using trigonometric functions like sine and cosine is essential for finding coordinates when angles are involved, especially for points on a circle.

Solve a trigonometric problem involving a circle with a radius of 8 units. Learn how to find the vertical distance between two points using polar coordinates and trigonometric functions. Suitable for high school students in Grades 10-12.

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