To find the function rule for g(theta), let's break down the given information and reasoning step by step:
1. We are given that both f(theta) and g(theta) are sine functions with f(0) = g(0) = 0. This means both functions start at the value of 0 when theta is 0.
2. It is also given that the amplitude of f(theta) is twice the amplitude of g(theta). The amplitude of a sine function is the distance from the midline to the maximum or minimum value. Let's denote the amplitude of g(theta) as A, then the amplitude of f(theta) would be 2A.
3. We are told that the period of f(theta) is one-half the period of g(theta). The period of a sine function is the length of one complete cycle, usually denoted as 2pi. So, the period of f(theta) would be (1/2) * 2pi = pi.
4. Lastly, it is given that f(pi/4) = 4. This means when theta is pi/4, the value of f(theta) is 4.
Now, based on this information, let's reason through the problem:
The standard form of a sine function f(theta) is f(theta) = A * sin(B(theta - C)) + D, where:
- A represents the amplitude, which is 2A for f(theta).
- B represents the frequency, which determines the period of the function.
- C represents the phase shift.
- D represents the vertical shift.
Since we're looking for the function rule for g(theta), we need to determine its amplitude, frequency, phase shift, and vertical shift.
1. Amplitude: We know that the amplitude of g(theta) is A. We are not given a specific value for A, so we cannot determine it directly from the information provided.
2. Frequency: We are told that the period of g(theta) is 2pi. The standard formula for frequency can be written as B = 2pi / period. Substituting the given period, we have B = 2pi / 2pi = 1.
3. Phase Shift: We are not given any information about the phase shift, so we assume there is no phase shift, which means C = 0.
4. Vertical Shift: We are not given any information about the vertical shift, so we assume there is no vertical shift, which means D = 0.
Putting it all together, we can write the function rule for g(theta) as:
g(theta) = A * sin(theta)
However, since we do not have a specific value for the amplitude A, we cannot determine the exact function rule for g(theta) based on the given information.