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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.

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If period of
f(\theta) is one-half the period of
g(\theta) and

g(\theta) has a period of 2π, then
T_(g) =2T_(f)=2 \pi and
T_(f)= \pi.

To find the period of sine function
f(\theta)=asin(b\theta+c) we use the rule
T_(f)= (2\pi)/(b).

f is sine function where f (0)=0, then c=0; with period
\pi, then
f(\theta)=asin 2\theta, because
T_(f)= (2 \pi )/(2) = \pi.

To find a we consider the condition
f( ( \pi )/(4) )=4, from where
asin2* (\pi)/(4) =a*sin ( \pi )/(2) =a=4.

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






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