Let's begin by graphing the functions $y = \sin(x)$ and $y = \cos(x)$ over one period. We'll also identify and label the amplitude, period, and key points on the graphs.

Graphs of $y = \sin(x)$ and $y = \cos(x)$

• Function: $y = \sin(x)$

• Amplitude: 1

• Period: $2\pi$

• Key Points:

  • $(0, 0)$
  • $\left(\frac{\pi}{2}, 1\right)$
  • $(\pi, 0)$
  • $\left(\frac{3\pi}{2}, -1\right)$
  • $(2\pi, 0)$

• Function: $y = \cos(x)$

• Amplitude: 1

• Period: $2\pi$

• Key Points:

  • $(0, 1)$
  • $\left(\frac{\pi}{2}, 0\right)$
  • $(\pi, -1)$
  • $\left(\frac{3\pi}{2}, 0\right)$
  • $(2\pi, 1)$

Let’s graph these functions.

Explanation of Amplitude and Period

• Amplitude: The amplitude of a wave is the maximum displacement of the wave from its mean position (the central axis). For the functions $y = \sin(x)$ and $y = \cos(x)$, the amplitude is 1, meaning the wave oscillates between 1 and -1. In the context of waves or oscillations, the amplitude represents the energy or intensity of the wave. A larger amplitude would indicate a more powerful wave (e.g., louder sound or brighter light).

• Period: The period of a wave is the distance (along the x-axis) required for the wave to complete one full cycle. For both $y = \sin(x)$ and $y = \cos(x)$, the period is $2\pi$. In wave or oscillation contexts, the period is the time it takes for one complete cycle of the wave to pass a fixed point. This could represent, for example, the time between successive peaks of a sound wave.

Would you like to see the graph or need any further explanation? Here are some questions to expand on this topic:

1. How would the graphs of $y = \sin(x)$ and $y = \cos(x)$ change if the amplitude were doubled?
2. What would happen to the graphs if the period were halved?
3. Can you describe the graph of $y = -\sin(x)$? How does it compare to $y = \sin(x)$?
4. How do the sine and cosine functions relate to circular motion?
5. What is the significance of phase shift in wave functions?

Tip: When working with trigonometric functions, always pay attention to the amplitude and period as they significantly affect the shape and behavior of the graph.