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].

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

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

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

  • $\sin(0) = 0$
  • $\sin\left(\frac{\pi}{2}\right) = 1$
  • $\sin(\pi) = 0$
  • $\sin\left(\frac{3\pi}{2}\right) = -1$
  • $\sin(2\pi) = 0$
  • Repeat these points for the second cycle up to $4\pi$.

• 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 $y = \cos(x)$

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

  • $\cos(0) = 1$
  • $\cos\left(\frac{\pi}{2}\right) = 0$
  • $\cos(\pi) = -1$
  • $\cos\left(\frac{3\pi}{2}\right) = 0$
  • $\cos(2\pi) = 1$
  • Repeat these points for the second cycle up to $4\pi$.

• 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 ($0$ to $4\pi$) and the y-values ($-1$ to $1$).
• 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.