tell me how to convert each degree measure into radians and each radian measure to degrees.

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To convert between degrees and radians, you need to understand that 180 degrees equals π radians. This is the fundamental relationship. To convert from degrees to radians, multiply the degree measure by π and divide by 180. For example, 30 degrees equals π divided by 6 radians. To convert from radians to degrees, multiply the radian measure by 180 and divide by π. The unit circle helps visualize these relationships, where a complete rotation is both 360 degrees and 2π radians. Let's look at some common degree to radian conversions. For 30 degrees, we multiply by π and divide by 180, giving us π over 6 radians, which is approximately 0.524 radians. For 45 degrees, we get π over 4 radians, or about 0.785 radians. 60 degrees converts to π over 3 radians, approximately 1.047 radians. A right angle of 90 degrees equals π over 2 radians, or about 1.571 radians. 180 degrees, a straight angle, equals π radians, which is approximately 3.142 radians. And a full circle of 360 degrees equals 2π radians, approximately 6.283 radians. Notice how the angle increases as we move around the circle. Now let's examine common radian to degree conversions. Pi over 6 radians equals 30 degrees. We multiply by 180 and divide by pi to get this result. Pi over 4 radians converts to 45 degrees. Pi over 3 radians equals 60 degrees. Pi over 2 radians is equivalent to 90 degrees, which is a right angle. Pi radians equals 180 degrees, representing a straight angle. And 2 pi radians equals 360 degrees, which is a full circle. As we move around the unit circle, we can see how these radian measures correspond to their degree equivalents. Let's solve some practical conversion problems. Problem 1: Convert 120 degrees to radians. We multiply 120 by π and divide by 180. This gives us 120π over 180, which simplifies to 2π over 3, or approximately 2.094 radians. Problem 2: Convert 3π over 4 radians to degrees. We multiply 3π over 4 by 180 over π. The π terms cancel out, leaving us with 3 times 180 over 4, which equals 540 over 4, or 135 degrees. On our number lines, we can see these conversions visually. 120 degrees corresponds to 2π over 3 radians, and 3π over 4 radians corresponds to 135 degrees. To summarize what we've learned about converting between degrees and radians: The fundamental relationship is that 180 degrees equals π radians. To convert from degrees to radians, multiply the degree measure by π and divide by 180. To convert from radians to degrees, multiply the radian measure by 180 and divide by π. It's helpful to memorize common values such as: 90 degrees equals π over 2 radians, 60 degrees equals π over 3 radians, 45 degrees equals π over 4 radians, and 30 degrees equals π over 6 radians. Remember that a full circle is 360 degrees or 2π radians. These conversions are essential in trigonometry, calculus, physics, and engineering applications.