Limits

To find the value of given that and are the roots of the quadratic equation , we will use the properties of roots and trigonometric identities.

1. Using Vieta's formulas, we know:
   • The sum of roots:
   • The product of roots:
2. We want to find . Using the trigonometric identity for sine of a sum:

1. Then, becomes:

1. Evaluating this within the options, the correct expression matches the first option:

Thus, the correct answer is .

To find the approximate value of , we can use the binomial approximation or the exponential function approximation for small values of .

First, let's use the exponential approximation:

For small values of , the expression can be approximated using:

where is the base of the natural logarithm, approximately equal to 2.718.

In our problem, and . So,

Now, calculate :

Hence, the approximate value of is 1.6.

This matches the correct answer given as 1.6.

To verify, note that this is consistent because the approximation for small values of works well here, as is very small.

Therefore, the correct answer is: 1.6.