Verified answer

(1) Since the minimum value of any sine = - 1, regardless of what the angle is called, the minimum value of that expression = - 19. (2) Unfortunately for your question, sin(pi/2) doesn't have a period - it is the fixed number pi/2 which is approx 1.57 .... But If you mean what is the period of y = sin(x/2) then since the period of y = sin x is 2pi, for sin(x/2) the period is 4pi.. the easiest way to see that is to consider y = sin2x, 3x, 4x, Ax. Then the periods are 2pi, pi, 2pi/3, pi/2, pi/A i.e. to find the period of y = sinAx just divide 2pi by A, so that if A = (1/2) we divide by the fraction (1/2) i.e. we multiply by 2/1 or just by 2 so for y = sin(x/2) the period = 2pi * 2 = 4pi. Similarly for (3): the period of y = tan x = pi, so the period of y = tan 2x = pi/2.

the minimum value of 19sin(theta) is -19

2 this is a constant it as no period

3 same as 2