๐ Understanding Sine and Tangent Functions
Sine and tangent functions are both trigonometric functions, but they behave very differently. The sine function, often denoted as $sin(x)$, represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. In contrast, the tangent function, $tan(x)$, is the ratio of the opposite side to the adjacent side. Let's explore their graphical representations and key characteristics.
๐ Definition of Sine Function
The sine function, $y = sin(x)$, oscillates between -1 and 1. Its graph is a smooth, continuous wave that repeats every $2\pi$ radians. The sine function starts at zero, increases to 1 at $\frac{\pi}{2}$, decreases to -1 at $\frac{3\pi}{2}$, and returns to zero at $2\pi$.
๐ Definition of Tangent Function
The tangent function, $y = tan(x)$, is defined as $sin(x)/cos(x)$. Unlike the sine function, the tangent function has vertical asymptotes where $cos(x) = 0$, specifically at $x = \frac{(2n+1)\pi}{2}$ for integer values of $n$. The tangent function's range is all real numbers, meaning it extends from negative infinity to positive infinity.
๐ Sine vs. Tangent: A Side-by-Side Comparison
| Feature |
Sine Function ($y = sin(x)$) |
Tangent Function ($y = tan(x)$) |
| Definition |
Ratio of opposite side to hypotenuse in a right triangle. |
Ratio of opposite side to adjacent side in a right triangle; also $sin(x)/cos(x)$. |
| Range |
$-1 \leq y \leq 1$ |
$-\infty < y < \infty$ |
| Period |
$2\pi$ |
$\pi$ |
| Continuity |
Continuous everywhere |
Discontinuous at $x = \frac{(2n+1)\pi}{2}$, where $n$ is an integer (vertical asymptotes). |
| Graph Shape |
Smooth, continuous wave |
Has vertical asymptotes and repeats more frequently than sine. |
| Symmetry |
Odd function (symmetric about the origin): $sin(-x) = -sin(x)$ |
Odd function (symmetric about the origin): $tan(-x) = -tan(x)$ |
| Zeros |
$x = n\pi$, where $n$ is an integer. |
$x = n\pi$, where $n$ is an integer. |
๐ก Key Takeaways
- ๐ Oscillation: The sine function's oscillation is bounded between -1 and 1, while the tangent function extends to infinity.
- ๐ Asymptotes: The tangent function possesses vertical asymptotes, making it discontinuous at certain points, unlike the continuous sine function.
- ๐ Periodicity: The sine function repeats every $2\pi$, whereas the tangent function repeats every $\pi$.