Question

Create a table of values to graph the function y = sine + 2/3

Answer 1

It looks like there might be a small error in the function you've provided. If you meant \( y = \sin(x) + \frac{2}{3} \), then I can certainly help you create a table of values for this function. The sine function oscillates between -1 and 1, so adding \(\frac{2}{3}\) will shift the entire graph upward.

Here’s a table of values for \( y = \sin(x) + \frac{2}{3} \) at standard intervals:

| \( x \) | \( \sin(x) \) | \( y = \sin(x) + \frac{2}{3} \) |
|-----------|------------------|-----------------------------------|
| \( 0 \) | \( 0 \) | \( \frac{2}{3} \approx 0.67 \) |
| \( \frac{\pi}{6} \) | \( \frac{1}{2} \) | \( \frac{2}{3} + \frac{1}{2} \approx 1.17 \) |
| \( \frac{\pi}{4} \) | \( \frac{\sqrt{2}}{2} \approx 0.707 \) | \( \frac{2}{3} + 0.707 \approx 1.37 \) |
| \( \frac{\pi}{3} \) | \( \frac{\sqrt{3}}{2} \approx 0.866 \) | \( \frac{2}{3} + 0.866 \approx 1.53 \) |
| \( \frac{\pi}{2} \) | \( 1 \) | \( \frac{2}{3} + 1 \approx 1.67 \) |
| \( \frac{2\pi}{3} \) | \( \frac{\sqrt{3}}{2} \approx 0.866 \) | \( \frac{2}{3} + 0.866 \approx 1.53 \) |
| \( \frac{3\pi}{4} \) | \( \frac{\sqrt{2}}{2} \approx 0.707 \) | \( \frac{2}{3} + 0.707 \approx 1.37 \) |
| \( \frac{5\pi}{6} \) | \( \frac{1}{2} \) | \( \frac{2}{3} + \frac{1}{2} \approx 1.17 \) |
| \( \pi \) | \( 0 \) | \( \frac{2}{3} \approx 0.67 \) |
| \( \frac{3\pi}{2} \) | \( -1 \) | \( \frac{2}{3} - 1 \approx -0.33 \) |
| \( 2\pi \) | \( 0 \) | \( \frac{2}{3} \approx 0.67 \) |

### Explanation:
- The table provides values for both \( x \) and \( y \) at key points in the interval from \( 0 \) to \( 2\pi \).
- Adjustments can be made to expand or reduce the range depending on your requirements.
- You can calculate additional values using a calculator or program if you want to plot more points.