The point \( \left(2, \frac{\pi}{4}\right) \) can also be represented by which of the following polar coordinates? Select all that apply. \( \square \) A. \( \left(2, \frac{3 \pi}{4}\right) \) B. \( \left(2,-\frac{\pi}{4}\right) \) \( \square \) C. \( \left(-2, \frac{5 \pi}{4}\right) \) \( \square \) D. \( \left(-2, \frac{9 \pi}{4}\right) \)

To determine which polar coordinates represent the same point as \( \left(2, \frac{\pi}{4}\right) \), we need to understand how polar coordinates work. The polar coordinate \( (r, \theta) \) represents a point in the plane where \( r \) is the distance from the origin and \( \theta \) is the angle measured from the positive x-axis. 1. **Basic Representation**: The point \( \left(2, \frac{\pi}{4}\right) \) means a point that is 2 units away from the origin at an angle of \( \frac{\pi}{4} \) radians. 2. **Negative Radius**: If the radius \( r \) is negative, the point is located in the opposite direction of the angle \( \theta \). Thus, \( (-r, \theta) \) corresponds to \( (r, \theta + \pi) \). 3. **Equivalent Angles**: Angles can be represented in multiple ways. For any angle \( \theta \), \( \theta + 2k\pi \) (where \( k \) is any integer) represents the same direction. Now, let's analyze each option: - **Option A**: \( \left(2, \frac{3\pi}{4}\right) \) - This point is 2 units away at an angle of \( \frac{3\pi}{4} \), which is in the second quadrant. This does not represent the same point as \( \left(2, \frac{\pi}{4}\right) \). - **Option B**: \( \left(2, -\frac{\pi}{4}\right) \) - This point is 2 units away at an angle of \( -\frac{\pi}{4} \), which is equivalent to \( \frac{7\pi}{4} \) (since \( -\frac{\pi}{4} + 2\pi = \frac{7\pi}{4} \)). This point is in the fourth quadrant, which does not represent the same point. - **Option C**: \( \left(-2, \frac{5\pi}{4}\right) \) - Here, the radius is negative. The angle \( \frac{5\pi}{4} \) is in the third quadrant. To find the equivalent positive radius, we can add \( \pi \) to the angle: \[ \left(-2, \frac{5\pi}{4}\right) \text{ is equivalent to } \left(2, \frac{5\pi}{4} + \pi\right) = \left(2, \frac{5\pi}{4} + \frac{4\pi}{4}\right) = \left(2, \frac{9\pi}{4}\right). \] Thus, this point represents the same location as \( \left(2, \frac{\pi}{4}\right) \). - **Option D**: \( \left(-2, \frac{9\pi}{4}\right) \) - Similar to option C, the angle \( \frac{9\pi}{4} \) can be simplified by subtracting \( 2\pi \): \[ \frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}. \] Thus, this point is equivalent to \( \left(2, \frac{\pi}{4}\right) \) as well. In conclusion, the options that represent the same point as \( \left(2, \frac{\pi}{4}\right) \) are: - \( \square \) C. \( \left(-2, \frac{5\pi}{4}\right) \) - \( \square \) D. \( \left(-2, \frac{9\pi}{4}\right) \)