complete the table for f(θ) for the given values of θ\n|θ|f(θ)|\n|0|\n|$\frac{pi}{4}$|\n|$\frac{pi}{2}$|\n|$\…

Explanation:

Step1: Calculate (f(\theta)) for (\theta = 0)

Substitute (\theta=0) into (f(\theta)=4\sin(2\theta)), we get (f(0)=4\sin(2\times0)=4\sin(0) = 0).

Step2: Calculate (f(\theta)) for (\theta=\frac{\pi}{4})

Substitute (\theta = \frac{\pi}{4}) into (f(\theta)=4\sin(2\theta)), then (2\theta=2\times\frac{\pi}{4}=\frac{\pi}{2}), and (f(\frac{\pi}{4})=4\sin(\frac{\pi}{2})=4\times1 = 4).

Step3: Calculate (f(\theta)) for (\theta=\frac{\pi}{2})

Substitute (\theta=\frac{\pi}{2}) into (f(\theta)=4\sin(2\theta)), so (2\theta = 2\times\frac{\pi}{2}=\pi), and (f(\frac{\pi}{2})=4\sin(\pi)=4\times0 = 0).

Step4: Calculate (f(\theta)) for (\theta=\frac{3\pi}{4})

Substitute (\theta=\frac{3\pi}{4}) into (f(\theta)=4\sin(2\theta)), then (2\theta=2\times\frac{3\pi}{4}=\frac{3\pi}{2}), and (f(\frac{3\pi}{4})=4\sin(\frac{3\pi}{2})=4\times(- 1)=-4).

Step5: Calculate (f(\theta)) for (\theta=\pi)

Substitute (\theta=\pi) into (f(\theta)=4\sin(2\theta)), so (2\theta=2\pi), and (f(\pi)=4\sin(2\pi)=4\times0 = 0).

Step6: Find the amplitude, mid - line and period

For a sine function of the form (y = A\sin(B\theta)+C), the amplitude is (|A|), the mid - line is (y = C) and the period is (T=\frac{2\pi}{|B|}). For (f(\theta)=4\sin(2\theta)), (A = 4), (B = 2) and (C = 0). The amplitude is (|4|=4), the mid - line is (y = 0) and the period is (T=\frac{2\pi}{2}=\pi).

Answer:

(\theta)	(f(\theta))
(0)	(0)
(\frac{\pi}{4})	(4)
(\frac{\pi}{2})	(0)
(\frac{3\pi}{4})	(-4)
(\pi)	(0)
The correct option for amplitude, mid - line and period is: amplitude = 4, midline: (y = 0), period=(\pi) (the third option).