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ketch and label the following graphs in your notebook. Get a tutor to check your graphs.1. 𝑦=sin(𝑥) over the domain [0,4𝜋].2. 𝑦=cos(𝑥) over the domain [0,4𝜋].

Question

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Solution

To sketch and label the graphs of the functions y=sin(x) y = \sin(x) and y=cos(x) y = \cos(x) over the domain [0,4π][0, 4\pi], follow these steps:

1. Graphing y=sin(x) y = \sin(x)

  • Identify key points: The sine function has a period of 2π 2\pi . Therefore, over the interval [0,4π][0, 4\pi], the sine function will complete two full cycles. Key points include:

    • sin(0)=0 \sin(0) = 0
    • sin(π2)=1 \sin\left(\frac{\pi}{2}\right) = 1
    • sin(π)=0 \sin(\pi) = 0
    • sin(3π2)=1 \sin\left(\frac{3\pi}{2}\right) = -1
    • sin(2π)=0 \sin(2\pi) = 0
    • Repeat these points for the second cycle up to 4π 4\pi .
  • Plot the points: Mark the key points on the graph and connect them smoothly to represent the wave-like nature of the sine function.

2. Graphing y=cos(x) y = \cos(x)

  • Identify key points: The cosine function also has a period of 2π 2\pi . Important points to consider over the interval [0,4π][0, 4\pi] include:

    • cos(0)=1 \cos(0) = 1
    • cos(π2)=0 \cos\left(\frac{\pi}{2}\right) = 0
    • cos(π)=1 \cos(\pi) = -1
    • cos(3π2)=0 \cos\left(\frac{3\pi}{2}\right) = 0
    • cos(2π)=1 \cos(2\pi) = 1
    • Repeat these points for the second cycle up to 4π 4\pi .
  • Plot the points: Similar to the sine function, mark the key points and connect them smoothly to show the behavior of the cosine function.

Tips for Graphing

  • Use graph paper if available, ensuring that both functions are drawn clearly.
  • Label each axis appropriately, including the values on the x-axis (00 to 4π4\pi) and the y-values (1-1 to 11).
  • Consider coloring each graph differently for distinction and clarity.

Once you have sketched the graphs, you can consult a tutor for feedback to ensure accuracy and clarity.

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