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To solve this problem, let's analyze what is being asked:

1. We have an angle, \( \theta = 235^\circ \).
2. We need to find:
   - A positive coterminal angle between \( 360^\circ \) and \( 720^\circ \).
   - A negative coterminal angle between \( -720^\circ \) and \( -360^\circ \).

### Step-by-Step Solution

#### Finding a Positive Coterminal Angle:
A coterminal angle can be found by adding or subtracting \( 360^\circ \) repeatedly to the given angle.

1. \( 235^\circ + 360^\circ = 595^\circ \)

   Since \( 595^\circ \) is between \( 360^\circ \) and \( 720^\circ \), it satisfies the condition for the positive coterminal angle.

#### Finding a Negative Coterminal Angle:
To find a negative coterminal angle, subtract \( 360^\circ \) from \( 235^\circ \) until the angle falls between \( -720^\circ \) and \( -360^\circ \).

1. \( 235^\circ - 360^\circ = -125^\circ \)

   Since \( -125^\circ \) is not between \( -720^\circ \) and \( -360^\circ \), it’s not the correct answer. Let’s subtract \( 360^\circ \) again.

2. \( -125^\circ - 360^\circ = -485^\circ \)

   Now, \( -485^\circ \) is between \( -720^\circ \) and \( -360^\circ \), so it satisfies the condition for the negative coterminal angle.

### Answer
The correct pair of angles is:
- Positive angle: \( 595^\circ \)
- Negative angle: \( -485^\circ \)

This matches the answer choice: **\( 595^\circ; -485^\circ \)**.

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Would you like more details or have any questions?

### Related Questions
1. How do we determine coterminal angles in general?
2. Why do we add or subtract \( 360^\circ \) to find coterminal angles?
3. What are coterminal angles used for in trigonometry?
4. How can you quickly find coterminal angles within a specific range?
5. Can this method be applied to radians as well?

### Tip
Remember that adding or subtracting multiples of \( 360^\circ \) (or \( 2\pi \) radians) to an angle will always yield a coterminal angle, meaning they share the same terminal side on the unit circle.

Find two angles, one positive between 360° and 720° and one negative between -720° and -360°, that are coterminal to θ = 235°.

Learn how to find a positive and negative coterminal angle for θ = 235°. This problem explores trigonometric concepts like coterminal angles, using formulas for both positive and negative angle ranges. Suitable for high school students.

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