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Consider the function f(theta)=3sin(2theta) where theta represents a number of radians. a. Complete the following table of values. theta f(theta) 0 0 (pi)/(4) (pi)/(2) (3pi)/(4) pi b. Graph the function f below.

Consider the function f(θ)=3sin⁡(2θ) f(\theta)=3 \sin (2 \theta) where θ \theta represents a number of radians.\newlinea. Complete the following table of values.\newline\begin{tabular}{|c|c|}\newline\hlineθ \theta & f(θ) f(\theta) \\\newline\hline 00 & 00 \\\newline\hlineπ4 \frac{\pi}{4} & \\\newline\hlineπ2 \frac{\pi}{2} & \\\newline\hline3π4 \frac{3 \pi}{4} & \\\newline\hlineπ \pi & \\\newline\hline\newline\end{tabular}\newlineb. Graph the function f f below.

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Full solution

Q. Consider the function f(θ)=3sin⁡(2θ) f(\theta)=3 \sin (2 \theta) where θ \theta represents a number of radians.\newlinea. Complete the following table of values.\newline\begin{tabular}{|c|c|}\newline\hlineθ \theta & f(θ) f(\theta) \\\newline\hline 00 & 00 \\\newline\hlineπ4 \frac{\pi}{4} & \\\newline\hlineπ2 \frac{\pi}{2} & \\\newline\hline3π4 \frac{3 \pi}{4} & \\\newline\hlineπ \pi & \\\newline\hline\newline\end{tabular}\newlineb. Graph the function f f below.
  1. Evaluate at 00: Evaluate f(θ)f(\theta) at θ=0\theta = 0. The function f(θ)=3sin⁡(2θ)f(\theta) = 3\sin(2\theta) requires us to plug in the value of θ\theta and multiply it by 22 before taking the sine. For θ=0\theta = 0, we have f(0)=3sin⁡(2×0)=3sin⁡(0)=0f(0) = 3\sin(2\times 0) = 3\sin(0) = 0.
  2. Evaluate at (π)/(4)(\pi)/(4): Evaluate f(θ)f(\theta) at θ=(π)/(4)\theta = (\pi)/(4). For θ=(π)/(4)\theta = (\pi)/(4), we have f((π)/(4))=3sin⁡(2∗(π)/(4))=3sin⁡((π)/(2))=3∗1=3f((\pi)/(4)) = 3\sin(2*(\pi)/(4)) = 3\sin((\pi)/(2)) = 3*1 = 3 since sin⁡((π)/(2))=1\sin((\pi)/(2)) = 1.
  3. Evaluate at (π)/(2)(\pi)/(2): Evaluate f(θ)f(\theta) at θ=(π)/(2)\theta = (\pi)/(2). For θ=(π)/(2)\theta = (\pi)/(2), we have f((π)/(2))=3sin⁡(2∗(π)/(2))=3sin⁡(π)=3∗0=0f((\pi)/(2)) = 3\sin(2*(\pi)/(2)) = 3\sin(\pi) = 3*0 = 0 since sin⁡(π)=0\sin(\pi) = 0.
  4. Evaluate at (3π)/(4)(3\pi)/(4): Evaluate f(θ)f(\theta) at θ=(3π)/(4)\theta = (3\pi)/(4). For θ=(3π)/(4)\theta = (3\pi)/(4), we have f((3π)/(4))=3sin⁡(2∗(3π)/(4))=3sin⁡((3π)/(2))=3∗(−1)=−3f((3\pi)/(4)) = 3\sin(2*(3\pi)/(4)) = 3\sin((3\pi)/(2)) = 3*(-1) = -3 since sin⁡((3π)/(2))=−1\sin((3\pi)/(2)) = -1.
  5. Evaluate at π\pi: Evaluate f(θ)f(\theta) at θ=π\theta = \pi. For θ=π\theta = \pi, we have f(π)=3sin⁡(2π)=3sin⁡(0)=0f(\pi) = 3\sin(2\pi) = 3\sin(0) = 0 since sin⁡(2π)=sin⁡(0)=0\sin(2\pi) = \sin(0) = 0.
  6. Complete the table: Complete the table with the evaluated values.\newlineThe completed table is:\newlineθ\theta | f(θ)f(\theta)\newline---------------------\newline00 | 00\newlineπ4\frac{\pi}{4} | 33\newlineπ2\frac{\pi}{2} | 00\newline3π4\frac{3\pi}{4} | −3-3\newlinef(θ)f(\theta)00 | 00
  7. Graph the function: Graph the function f(θ)=3sin⁡(2θ)f(\theta) = 3\sin(2\theta). To graph the function, plot the points from the table on a coordinate system with θ\theta on the horizontal axis and f(θ)f(\theta) on the vertical axis. Then, draw a smooth curve through the points, keeping in mind the periodic nature of the sine function with a period of π\pi for 3sin⁡(2θ)3\sin(2\theta).

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