To determine which trigonometric value is larger without a calculator, one can analyze the angle's quadrant. For an angle of 2 radians, which is approximately 114.6 degrees, it lies in the second quadrant where sine is positive and cosine is negative. Thus, it can be concluded that sin 2 is greater than cos 2. The discussion highlights the importance of understanding the unit circle and quadrants in trigonometry. Additionally, the user reflects on their educational experience and plans to study trigonometry more thoroughly.