On the unit circle, $\tan(\theta)$ represents the slope of the terminal ray. It's the ratio of the y-coordinate (sine) to the x-coordinate (cosine).

Tangent has vertical asymptotes where $\cos(\theta) = 0$, because $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Division by zero is undefined.

The slope of the terminal ray on the unit circle repeats every $\pi$ radians (half-rotation), unlike sine and cosine which repeat every $2\pi$ radians.

Quadrant I: Positive (both sine and cosine are positive). Quadrant II: Negative (sine is positive, cosine is negative). Quadrant III: Positive (both sine and cosine are negative). Quadrant IV: Negative (sine is negative, cosine is positive).

As $x$ approaches an asymptote, $\tan(x)$ approaches either positive infinity or negative infinity. The function becomes infinitely steep.

A negative 'a' value reflects the graph of $y = \tan(x)$ over the x-axis.

Changing 'b' in $y = \tan(bx)$ alters the period of the tangent function. A larger 'b' compresses the graph horizontally, decreasing the period, while a smaller 'b' stretches the graph horizontally, increasing the period.

The range of the standard tangent function, $y = \tan(x)$, is all real numbers, or $(-\infty, \infty)$.

Phase shift is the horizontal translation of a tangent function. In the equation $y = a \tan(b(x - c)) + d$, 'c' represents the phase shift. A positive 'c' shifts the graph to the right, while a negative 'c' shifts it to the left.

The 'd' value represents the vertical shift of the tangent function. A positive 'd' shifts the graph upward, while a negative 'd' shifts it downward.

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$

A vertical line $x = a$ where the function approaches infinity or negative infinity as $x$ approaches $a$.

The smallest positive value $P$ such that $f(x + P) = f(x)$ for all $x$ in the domain of $f$.

A horizontal translation of a function's graph.

A transformation that stretches or compresses a graph vertically.

The set of all possible output values (y-values) of a function.

A transformation that flips the graph of a function over the x-axis, changing the sign of the y-values.

A circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system.

Functions that relate the angles of a triangle to the ratios of its sides.

The set of all possible input values (x-values) for which the function is defined.

Tangent: Vertical asymptotes, period of $\pi$, range of $(-\infty, \infty)$. Sine: No asymptotes, period of $2\pi$, range of $[-1, 1]$.

Tangent: Period is $\pi$. Cosine: Period is $2\pi$.

Tangent: Range is $(-\infty, \infty)$. Sine: Range is $[-1, 1]$.

Period changes are calculated differently. Tangent: $T = \frac{\pi}{|b|}$. Sine: $T = \frac{2\pi}{|b|}$. Asymptotes exist for tangent but not for sine.

Tangent: Asymptotes at $x = \frac{\pi}{2} + k\pi$. Cotangent: Asymptotes at $x = k\pi$ (where k is an integer).

As $x$ approaches 0, $\tan(x)$ approaches 0. As $x$ approaches 0, $\sin(x)$ approaches 0.

Tangent: Odd function, symmetric about the origin. Cosine: Even function, symmetric about the y-axis.

Tangent: All real numbers except $x = \frac{\pi}{2} + k\pi$. Sine: All real numbers.

Tangent: Increasing between asymptotes. Cotangent: Decreasing between asymptotes. Asymptotes are located in different places.

Tangent: Vertical stretch/compression, reflection if 'a' is negative. Sine: Amplitude change, reflection if 'a' is negative.