Question

Sinx graph

Answer 1

The graph of $y = \sin x$ is a continuous wave with a domain of all real numbers and a range of $[-1, 1]$. It has a period of $2\pi$, repeating its pattern every $2\pi$ units. Key points include $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, and $(2\pi, 0)$.

Answer 2

The graph of $y = \sin x$ is a fundamental trigonometric function characterized by its wave-like shape.

1. Domain: The sine function is defined for all real numbers, so its domain is $\mathbb{R}$ (all real numbers).
2. Range: The output values of the sine function oscillate between -1 and 1, inclusive. Thus, the range is $[-1, 1]$.
3. Periodicity: The sine function is periodic with a period of $2\pi$. This means the graph repeats its pattern every $2\pi$ units along the x-axis.
4. Key Points for One Cycle (0 to $2\pi$):
   • At $x=0$, $\sin(0) = 0$. The graph passes through the origin $(0,0)$.
   • At $x=\frac{\pi}{2}$, $\sin(\frac{\pi}{2}) = 1$. This is the maximum value, occurring at $(\frac{\pi}{2}, 1)$.
   • At $x=\pi$, $\sin(\pi) = 0$. The graph crosses the x-axis at $(\pi, 0)$.
   • At $x=\frac{3\pi}{2}$, $\sin(\frac{3\pi}{2}) = -1$. This is the minimum value, occurring at $(\frac{3\pi}{2}, -1)$.
   • At $x=2\pi$, $\sin(2\pi) = 0$. The graph completes one cycle and returns to the x-axis at $(2\pi, 0)$.

The graph is a smooth, continuous curve that connects these key points and extends infinitely in both the positive and negative x-directions, repeating the same wave pattern.