Final answer:
The measure of the acute angle a in the right triangle with a hypotenuse of 11.6 and an opposite side of 9 can be found by using the sine function and calculating sin-1(9 / 11.6), which is approximately 53 degrees.
Step-by-step explanation:
To determine the measure of the acute angle a in the given right triangle when the hypotenuse is 11.6 and the opposite side is 9, we can use the sine function (sin) which relates the opposite side to the hypotenuse in a right triangle. The formula sin(a) = opposite/hypotenuse can be applied here to find the acute angle a.
So, sin(a) = 9 / 11.6. Using a calculator, we find that a ≈ sin-1(9 / 11.6). After calculating this, we get that angle a is approximately 53 degrees, which is the measure of the acute angle in the triangle.
The Pythagorean theorem could also be referenced here to check the lengths of the sides of the triangle. However, since we are only looking for the angle and have the length of the opposite side and the hypotenuse, it is not needed in this case.