As x increases from p/4 to 3p/2, the value of sin(x) :

A.increases throughout the interval
B.decreases at first, then increases
C.increases at first, then decreases
D.decreases throughout the interval
E.none of the above

please explain why the answer is as it is, I don't understand this at all, please help :)

This only makes sense if your "p" is pi, and x is an angle expressed in radians.

The sin of pi/4 is 0.7071
Then sin x rises to +1 at x = pi/2, and falls to -1 at x = 3 pi/2

The correct answer is therefore C

To determine how the value of sin(x) changes as x increases from p/4 to 3p/2, we can look at the graph of the sine function.

The sine function is a periodic function that oscillates between -1 and 1 as x increases. It starts at 0 when x = 0, reaches a maximum value of 1 when x = p/2, returns to 0 when x = p, reaches a minimum value of -1 when x = 3p/2, and returns to 0 again when x = 2p.

In this particular problem, we are interested in the interval from p/4 to 3p/2. Note that p/4 is less than p/2, which means that x starts in the first quadrant where sin(x) is positive. As x increases, it goes through the second quadrant where sin(x) is still positive until it reaches p/2. At this point, sin(x) is at its maximum value of 1.

After p/2, x continues to increase and enters the third quadrant where sin(x) becomes negative. The value of sin(x) starts decreasing and reaches 0 when x = p. So, from p/2 to p, sin(x) decreases.

Finally, as x continues to increase from p to 3p/2, it enters the fourth quadrant where sin(x) remains negative until it reaches 3p/2. At this point, sin(x) is at its minimum value of -1.

So, to summarize, as x increases from p/4 to 3p/2, the value of sin(x) increases at first (from p/4 to p/2), then decreases (from p/2 to p), and finally decreases (from p to 3p/2).

Therefore, the correct answer is option C: increases at first, then decreases.