The functions f(θ) and g(θ) are sine functions, where f(0) = g(0) = 0.

The amplitude of f(θ) is twice the amplitude of g(θ). The period of f(θ) is one-half the period of g(θ). If g(θ) has a period of 2pi and f (pi/4) = 4, write the function rule for g(θ).

g(θ) has a period of 2π means:

g(θ) = a sin(θ)

The amplitude of f(θ) is twice the amplitude of g(θ).
The period of f(θ) is one-half the period of g(θ).
means:
f(θ) = 2a sin(2θ)

f(π/4) = 2a sin(π/2) = 2a = 4, so a=2
g(θ) = 2sin(θ)

To find the function rule for g(θ), we need to consider the given information:

1. The period of g(θ) is 2π.
2. f(θ) has a period one-half the period of g(θ).
3. The amplitude of f(θ) is twice the amplitude of g(θ).
4. f(pi/4) = 4.

Let's break down the steps to find the function rule for g(θ):

Step 1: Determine the amplitude of g(θ).
Since the amplitude of f(θ) is twice the amplitude of g(θ), and we know that f(pi/4) = 4, we can deduce that the amplitude of f(θ) is 4.

Step 2: Determine the amplitude of g(θ).
Since the amplitude of g(θ) is half the amplitude of f(θ), we divide the amplitude of f(θ) by 2. Therefore, the amplitude of g(θ) is 4/2 = 2.

Step 3: Determine the function rule for g(θ).
To find the function rule for g(θ), we need to consider the general form of a sine function: y = A*sin(B(x - C)) + D, where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.

Since we have determined the amplitude of g(θ) is 2 and the period of g(θ) is 2π, we can substitute these values into the general form of the sine function to get g(θ).

Therefore, the function rule for g(θ) is: g(θ) = 2*sin((2π/2π)(θ - C)) + D = 2*sin(θ - C) + D

Step 4: Use additional information to determine the values of C and D.
We know that g(0) = 0. Substituting θ = 0 into the function rule, we get: 0 = 2*sin(0 - C) + D.

Since sin(0) = 0, we have 0 = 2*0 + D, which simplifies to D = 0.

Step 5: Determine the value of C using the given period of f(θ) and g(θ).
We know that the period of g(θ) is given as 2π. Since the period of f(θ) is one-half the period of g(θ), the period of f(θ) is π.

We also know that f(pi/4) = 4. Substituting θ = pi/4 into the function rule for f(θ), we get: 4 = 2*sin((π/π)*(pi/4 - C)) + 0.

Simplifying, we have 4 = 2*sin(pi/4 - C). Divide both sides by 2 to isolate sin(pi/4 - C): 2 = sin(pi/4 - C).

To determine the value of C, we can take the inverse sine of both sides: pi/4 - C = arcsin(2).

Rearranging the equation, C = pi/4 - arcsin(2).

Therefore, the function rule for g(θ) is: g(θ) = 2*sin(θ - (pi/4 - arcsin(2))).

In summary, the function rule for g(θ) is g(θ) = 2*sin(θ - (pi/4 - arcsin(2))).