The functions f (theta) and g (theta) are sine functions, where f (0) equals g (0) equals 0.the amplitude of f (theta) is twice the amplitude of g (theta). the period of f (theta) is one-half the period of g (theta). if g (theta) has a period of 2pi and f(pi/4)=4, write the function rule for g (theta). explain your reasoning.

If period of is one-half the period of and
  has a period of 2π, then and .

To find the period of sine function we use the rule .

f is sine function where f (0)=0, then c=0; with period , then , because . 

To find a we consider the condition , from where .

If the amplitude of is twice the amplitude of , then has a product factor twice smaller than and while period of is 2π and g(0)=0, we can write .