Graphs and Basic Properties of Sine and Cosine
1. Building the Graph of $y = \sin x$
Step 1: Compute Key Values on $[-\pi, \pi]$
We begin by evaluating $\sin x$ at key angles in the interval $[-\pi, \pi]$:
| $x$ | $-\pi$ | $-\frac{3\pi}{4}$ | $-\frac{\pi}{2}$ | $-\frac{\pi}{4}$ | $0$ | $\frac{\pi}{4}$ | $\frac{\pi}{2}$ | $\frac{3\pi}{4}$ | $\pi$ |
|---|---|---|---|---|---|---|---|---|---|
| $\sin x$ | $0$ | $-\frac{\sqrt{2}}{2} \approx -0.71$ | $-1$ | $-\frac{\sqrt{2}}{2} \approx -0.71$ | $0$ | $\frac{\sqrt{2}}{2} \approx 0.71$ | $1$ | $\frac{\sqrt{2}}{2} \approx 0.71$ | $0$ |
Step 2: Plot Points and Connect with a Smooth Curve
Plot each $(x, \sin x)$ pair on the coordinate plane and connect them with a smooth, wave-like curve — no sharp corners!
Step 3: Extend the Curve Using Periodicity
Since $\sin x$ repeats every $2\pi$, we shift the curve we drew on $[-\pi, \pi]$ by $\pm 2\pi$, $\pm 4\pi$, $\pm 6\pi$, and so on in both directions. This produces the full sinusoid extending infinitely left and right.
Key Fact: The graph of $y = \sin x$ is called a sinusoid. It is a smooth, continuous wave that stays entirely within the horizontal strip between $y = -1$ and $y = 1$.
Interactive Graph of $y = \sin x$
Interactive visualization — available in the full lesson
2. Properties of $y = \sin x$
- Domain: $(-\infty, +\infty)$ — all real numbers
-
Zeros: $x = \pi n$, where $n \in \mathbb{Z}$
i.e., $\ldots, -2\pi, -\pi, 0, \pi, 2\pi, 3\pi, \ldots$ -
Positive: on intervals $(2\pi n,\; \pi + 2\pi n)$, $n \in \mathbb{Z}$
Negative: on intervals $(\pi + 2\pi n,\; 2\pi + 2\pi n)$, $n \in \mathbb{Z}$ -
Symmetry: Odd function
$\sin(-x) = -\sin x$ — graph is symmetric about the origin
-
Increasing: on $\left[-\frac{\pi}{2} + 2\pi n,\; \frac{\pi}{2} + 2\pi n\right]$, $n \in \mathbb{Z}$
Decreasing: on $\left[\frac{\pi}{2} + 2\pi n,\; \frac{3\pi}{2} + 2\pi n\right]$, $n \in \mathbb{Z}$ -
Period: $T = 2\pi$ (fundamental period)
$\sin(x + 2\pi) = \sin x$ for all $x$ - Range: $[-1, 1]$ — the function is bounded
-
Maximum: $\sin x = 1$ at $x = \frac{\pi}{2} + 2\pi n$
Minimum: $\sin x = -1$ at $x = -\frac{\pi}{2} + 2\pi n$ - Not invertible: Any horizontal line $y = c$ where $-1 \le c \le 1$ intersects the graph infinitely many times