Okay, let's see. I need to compute the arcsin of -1/2 and express the answer in radians. Hmm, arcsin is the inverse sine function, right? So it's asking for the angle whose sine is -1/2. Let me recall the unit circle and the values of sine for common angles.

First, I remember that sine is negative in the third and fourth quadrants. The arcsin function, however, has a range of [-π/2, π/2], which corresponds to angles in the fourth and third quadrants but only from -π/2 up to π/2. Wait, no. Actually, the range of arcsin is [-π/2, π/2], so it can only give angles in the fourth and first quadrants. But since we're dealing with a negative value here, -1/2, the angle should be in the fourth quadrant because sine is negative there. But wait, the range of arcsin is from -π/2 to π/2, so the fourth quadrant is from -π/2 to 0, and the first quadrant is from 0 to π/2. So the answer should be a negative angle between -π/2 and 0.

Now, let's think about the reference angle. The sine of π/6 is 1/2, so the reference angle here is π/6. But since the sine is negative, the angle should be -π/6. Let me verify that. If I take sin(-π/6), that's equal to -sin(π/6) = -1/2. Yes, that's correct. So arcsin(-1/2) should be -π/6.

Wait, but sometimes I might get confused with the quadrants. Let me double-check. The sine of π/6 is 1/2, so the reference angle is π/6. Since we're dealing with a negative value, and the range of arcsin is [-π/2, π/2], the angle must be -π/6. That makes sense. So the answer is -π/6 radians.

But just to make sure I didn't mix up the quadrants. The arcsin function returns angles in the range of -π/2 to π/2, which covers the fourth and first quadrants. Since the input is negative, the angle must be in the fourth quadrant, which is between -π/2 and 0. The reference angle for 1/2 is π/6, so the angle in the fourth quadrant with a sine of -1/2 is -π/6. That seems right.

Another way to think about it is to consider the unit circle. Starting from the positive x-axis, moving clockwise (since the angle is negative) by π/6 radians would land us at -π/6, which has a sine value of -1/2. Yes, that checks out.

I don't think I made any mistakes here. The key steps are recognizing that arcsin returns angles in a specific range and that the negative sine value implies the angle is in the fourth quadrant. The reference angle is π/6, so the answer is just -π/6.
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