How do you graph y = sin(x)?

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To graph \( y = \sin(x) \), plot the sine values for \( x \) from \( 0 \) to \( 2\pi \) and connect the points.

To start, understand that the sine function, \( \sin(x) \), is a periodic function with a period of \( 2\pi \). This means that the graph repeats every \( 2\pi \) units along the \( x \)-axis. The basic shape of the sine curve is a smooth, wave-like pattern that oscillates above and below the \( x \)-axis.

First, create a set of axes with the \( x \)-axis ranging from \( 0 \) to \( 2\pi \) (approximately 6.28) and the \( y \)-axis ranging from -1 to 1. These ranges are chosen because \( \sin(x) \) values always lie between -1 and 1.

Next, plot key points where the sine function has known values:
- At \( x = 0 \), \( \sin(0) = 0 \).
- At \( x = \frac{\pi}{2} \) (approximately 1.57), \( \sin\left(\frac{\pi}{2}\right) = 1 \).
- At \( x = \pi \) (approximately 3.14), \( \sin(\pi) = 0 \).
- At \( x = \frac{3\pi}{2} \) (approximately 4.71), \( \sin\left(\frac{3\pi}{2}\right) = -1 \).
- At \( x = 2\pi \), \( \sin(2\pi) = 0 \).

After plotting these points, draw a smooth curve through them. The curve should start at the origin (0,0), rise to (π/2, 1), fall back to (π, 0), continue down to (3π/2, -1), and finally return to (2π, 0). This completes one full cycle of the sine wave.

For a more detailed graph, you can plot additional points between these key points, such as \( x = \frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \), and calculate their sine values. This will help you draw a more accurate curve. Remember,