Graphs of Sine/Cosine Functions

Overview

The sine and cosine functions, $\sin(x)$ and $\cos(x)$, are fundamental periodic functions in trigonometry. Their graphs, often referred to as sine waves and cosine waves, are used extensively in various fields such as physics, engineering, and signal processing.

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Key Properties of $\sin(x)$ and $\cos(x)$

Periodicity

Both $\sin(x)$ and $\cos(x)$ are periodic with a period of $2\pi$

$$\sin(x + 2\pi) = \sin(x)$$ $$\cos(x + 2\pi) = \cos(x)$$

Amplitude

The amplitude (maximum absolute value) is 1 for both functions.

$$-1 \leq \sin(x) \leq 1$$ $$-1 \leq \cos(x) \leq 1$$

Domain, Range & Period

• Domain: All real numbers $x$
• Range: $[-1, 1]$
• Period: $2\pi$ radians (360°)

Key Points

• $\sin(x)$ starts at $(0, 0)$, rises to $1$ at $\frac{\pi}{2}$, and returns to $0$ at $\pi$
• $\cos(x)$ starts at $(0, 1)$, drops to $0$ at $\frac{\pi}{2}$, and reaches $-1$ at $\pi$

Symmetry

$\sin(x)$ is an odd function:

$$\sin(-x) = -\sin(x)$$

$\cos(x)$ is an even function:

$$\cos(-x) = \cos(x)$$

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Graph Characteristics

Sine Function $\sin(x)$

• Starts at $(0, 0)$
• Crosses the $x-axis$ at multiples of $\pi$

$$x = 0, \pi, 2\pi, \dots$$

• Peaks at $\frac{\pi}{2}$ and valleys at $\frac{3\pi}{2}$

Cosine Function $\cos(x)$

• Starts at $(0, 1)$
• Crosses the $x-axis$ at odd multiples of $\frac{\pi}{2}$

$$x = \frac{\pi}{2}, \frac{3\pi}{2}, \dots$$

• Peaks at $x = 0, 2\pi, \dots$, and valleys at $x = \pi, 3\pi, \dots$

Shifts and Scaling

General forms:

$$f(x) = a + b \sin(cx) \quad \text{or} \quad g(x) = a + b \cos(cx)$$

where

• $a$ adjusts the vertical shift
• $b$ adjusts the amplitude
• $c$ adjusts the frequency.

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