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To convert polar coordinates \((-8, -222^\circ)\) to rectangular (Cartesian) coordinates, we use the formulas:

\[
x = r \cos \theta
\]
\[
y = r \sin \theta
\]

where:
- \( r = -8 \)
- \( \theta = -222^\circ \)

### Step 1: Converting the Angle
Since the angle is negative, we can convert it to a positive equivalent by adding 360°:
\[
-222^\circ + 360^\circ = 138^\circ
\]

Thus, we can use \( \theta = 138^\circ \).

### Step 2: Calculating \( x \) and \( y \)
1. **Calculate \( x \):**
   \[
   x = -8 \cos(138^\circ)
   \]

2. **Calculate \( y \):**
   \[
   y = -8 \sin(138^\circ)
   \]

Now, evaluating these:

1. \(\cos(138^\circ) \approx -0.7431\), so:
   \[
   x = -8 \times -0.7431 = 5.9448
   \]

2. \(\sin(138^\circ) \approx 0.6691\), so:
   \[
   y = -8 \times 0.6691 = -5.3528
   \]

### Final Answer
The rectangular coordinates are approximately:
\[
(x, y) \approx (5.9448, -5.3528)
\]

Would you like further details or have any questions?

### Further Questions to Explore
1. How can we convert angles in radians instead of degrees?
2. What changes if the radius \( r \) were positive?
3. How does the quadrant affect the final signs of \( x \) and \( y \)?
4. Can we convert from rectangular back to polar coordinates?
5. What is the impact of other values of \( r \) or \( \theta \)?

### Tip
Always ensure the angle is in the correct quadrant by adjusting with 360° if the angle is negative.

Convert -8∠-222° to rectangular coordinates.

Learn how to convert polar coordinates (-8, -222°) to rectangular coordinates with a step-by-step solution. This problem involves trigonometric functions and polar-to-rectangular coordinate conversion, suitable for Grades 9-10.

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