\(\sin ^{2}(4 \pi / 3)+\sin (\pi / 6)\) then \(A=\) (a) \((3 / 4)\) (b) \((5 / 4)\) (c) \((5 / 2)\) (d) \((4 / 5)\)

First, we need to determine the value of \(\sin(4\pi/3)\). Recall that the sine function has a period of \(2\pi\), so we can find an equivalent angle in the first rotation by subtracting \(2\pi\) from the given angle: \[\sin(4\pi/3 - 2\pi) = \sin(4\pi/3 - 6\pi/3) = \sin(-2\pi/3)\] Now we have an angle in the second rotation, and we can use the property that the sine function is odd, i.e., \(\sin(-x) = -\sin(x)\), to find the value of the function at this angle: \[\sin(-2\pi/3) = -\sin(2\pi/3)\] The angle \(2\pi/3\) is in the second quadrant, so we can use the fact that \(\sin(\pi - x) = \sin(x)\) for angles in the second quadrant: \[-\sin(2\pi/3) = -\sin(\pi - \pi/3) = -\sin(\pi/3)\] Finally, we know that \(\sin(\pi/3) = \frac{\sqrt{3}}{2}\), so we have: \[-\sin(4\pi/3) = -\frac{\sqrt{3}}{2}\]

Now that we have \(\sin(4\pi/3)\), we can now find its square: \[\sin^2(4\pi/3) = \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\]

We know that \(\sin(\pi/6) = \frac{1}{2}\) from the basic trigonometric values.

Now that we have both values, we can find their sum: \[\sin^2(4\pi/3) + \sin(\pi/6) = \frac{3}{4} + \frac{1}{2}\]

To add the two fractions, we need to find a common denominator. The lowest common denominator for 4 and 2 is 4. Rewrite the second fraction with a denominator of 4: \[\frac{3}{4} + \frac{1\cdot 2}{2\cdot 2} = \frac{3}{4} + \frac{2}{4}\] Now add the two fractions: \[\frac{3}{4} + \frac{2}{4} = \frac{3+2}{4} = \frac{5}{4}\] The sum is \(\frac{5}{4}\).

Given the sum \(\frac{5}{4}\), we can identify the correct answer as option (b). So, \(A = \frac{5}{4}\).