Working with a sinusoidal function

(1)
our function is

y/3 = 3+ 4sin [3x+ pi ]

when x=pi/2

We have, 3x + pi = (3pi/2)+pi = 5pi/2 = (2pi + pi/2)

Sin function repeats itself when we add 2pi to it

Hence Sin(2pi + pi/2)= Sin(pi/2) = 1

Substitute this value back to the given function and find the value of y.

y/3 = 3+ 4sin [3x+ pi ] Gives the answer y = 21

Case1

y = {3+4(sin[3x+ pi])/3} Gives the answer y = 5
Because Sin(5pi/6)= 0.5 )

Case2

y = 3+4 Sin[3x+(pi/3)] Gives the answer y = 1 (One among your ans)

Sin (3x+pi/3)= Sin(3pi/2 + pi/3) = Sin(11pi/6) = -0.5

y = 3+ (4 x -0.5 ) = 3-2= 1

(2)
A periodic function can be defined as any function for which,

f(x) = f (x + T) ......(i) for all x.

(that is, value of the function is the same after we add another constant (T) to it)

The smallest constant T which satisfies the above equation is called the period (or the fundamental period) of the function.

The period of the function y = sin Ax is 2pi/A.

Note that A = | A |

The function's value must remain invariant as we use x in it. The smallest value of x for which the function is invariant is the period.

This gives, x = 2(pi/3) (Answer is F)

We need not consider other values as the only variable is x.

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