Evaluate each of the following:

• (i) sin−1(sin 12)
• (ii) sin−1(sin 2)

To evaluate the expressions sin^-1(sin 12) and sin^-1(sin 2), we need to consider the properties of the sine function and its inverse.

Evaluating sin^-1(sin 12)

The sine function is periodic with a period of 2π (approximately 6.28). The range of the inverse sine function, sin^-1, is limited to [-π/2, π/2] (or approximately [-1.57, 1.57]).

First, we convert 12 radians to a value within the range of sin^-1:

• 12 radians is greater than π, so we can subtract 2π to find an equivalent angle:
• 12 - 2π ≈ 12 - 6.28 = 5.72 radians.
• 5.72 is still outside the range, so we subtract another 2π:
• 5.72 - 2π ≈ 5.72 - 6.28 = -0.56 radians.

Now, we find sin^-1(sin 12):

Since -0.56 is within the range of sin^-1, we have:

sin^-1(sin 12) = -0.56 radians.

Evaluating sin^-1(sin 2)

Next, we look at sin^-1(sin 2). Since 2 radians is already within the range of [-π/2, π/2], we can directly evaluate:

sin^-1(sin 2) = 2 radians.

Summary

In summary:

• sin^-1(sin 12) = -0.56 radians
• sin^-1(sin 2) = 2 radians