To find the slope of the tangent line to the sine function at the origin (0), we need to determine the derivative of the sine function at that point. The derivative of the sine function is the cosine function:
f(x) = sin(x)
f'(x) = cos(x)
At the origin, we evaluate the derivative as follows:
f'(0) = cos(0) = 1
This means that the slope of the tangent line to the sine function at the origin is 1.
Now, let’s consider the interval from 0 to 2π. In this interval, the sine function completes one full cycle. The sine function starts at 0, increases to 1 at π/2, decreases back to 0 at π, goes down to -1 at 3π/2, and returns to 0 at 2π. Thus, we see that:
- The number of complete cycles of the sine function from 0 to 2π is 1.
Conclusion: We have found that the slope of the tangent line to the sine function at the origin is 1, while there is 1 complete cycle in the interval from 0 to 2π. This suggests that at the beginning of the sine wave, the function is increasing at a consistent rate, which aligns with the single completed cycle observed in that interval.