circle subtended by the angle. (F.TF.A.1,F.TF.A.2) 1. The angle \( \frac{\pi}{4} \) radians has the same sine value as which of the following angles? Give an explanation of your answer.
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The angle \( \frac{\pi}{4} \) radians, which is equivalent to 45 degrees, has the same sine value as \( \frac{3\pi}{4} \) radians (or 135 degrees) and \( \frac{5\pi}{4} \) radians (or 225 degrees). This is because of the symmetry of the sine function in the unit circle, where sine is positive in the first and second quadrants and negative in the third and fourth quadrants. Specifically, the sine values of angles in the first and second quadrants mirror each other. Similarly, the angles \( 2\pi - \frac{\pi}{4} \) or \( 2\pi + \frac{\pi}{4} \), cyclically bringing you back to the same sine value, also demonstrate this property. Thus, there are infinitely many angles sharing the sine value of \( \frac{\pi}{4} \) based on periodicity!