If period of [tex]f(\theta)[/tex] is one-half the period of [tex]g(\theta)[/tex] and
[tex]g(\theta)[/tex] has a period of 2π, then [tex]T_{g} =2T_{f}=2 \pi [/tex] and [tex]T_{f}= \pi [/tex].
To find the period of sine function [tex]f(\theta)=asin(b\theta+c)[/tex] we use the rule [tex]T_{f}= \frac{2\pi}{b} [/tex].
f is sine function where f (0)=0, then c=0; with period [tex] \pi [/tex], then [tex]f(\theta)=asin 2\theta[/tex], because [tex]T_{f}= \frac{2 \pi }{2} = \pi [/tex].
To find a we consider the condition [tex]f( \frac{ \pi }{4} )=4[/tex], from where [tex]asin2* \frac{\pi}{4} =a*sin \frac{ \pi }{2} =a=4[/tex].
If the amplitude of [tex]f(\theta)[/tex] is twice the amplitude of [tex]g(\theta)[/tex] , then [tex]g(\theta)[/tex] has a product factor twice smaller than [tex]f(\theta)[/tex] and while period of [tex]g(\theta)[/tex] is 2π and g(0)=0, we can write [tex]g(\theta)=2sin\theta[/tex].
