8.2 Graphs of the Other Trigonometric Functions

Graphs of the Other Trigonometric Functions

Tangent, cotangent, secant, and cosecant each behave differently from sine and cosine. They introduce vertical asymptotes, different periods, and shapes that can catch you off guard if you only studied sine and cosine graphs. This section covers what each graph looks like, where the asymptotes and intercepts fall, and how transformations change them.

Graphs of Tangent, Secant, Cosecant, and Cotangent Functions

Graph of the Tangent Function

The tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$. Because of that fraction, the graph behaves very differently from sine or cosine. Wherever $\cos x = 0$, the function is undefined, and you get a vertical asymptote.

• Period: $\pi$ (the pattern repeats every $\pi$ units, not $2\pi$ like sine and cosine)
• Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer
• $x$-intercepts: $x = n\pi$, where $n$ is any integer
• The graph passes through the origin $(0, 0)$
• Domain: All real numbers except the asymptote locations
• Range: All real numbers

Between each pair of consecutive asymptotes, the graph rises from $-\infty$ to $+\infty$. It's an odd function, meaning $\tan(-x) = -\tan(x)$, so the graph has rotational symmetry about the origin.

A common mistake: students sometimes think tangent has an amplitude. It doesn't. The graph extends infinitely in both the positive and negative $y$-directions within each period.

Variations of Tangent Functions

The general form is $y = a\tan(b(x - h)) + k$. Here's what each parameter does:

• Vertical stretch/compression ($a$): $y = a\tan x$ stretches the graph vertically by a factor of $|a|$. If $|a| > 1$, the graph is steeper near the intercepts. If $0 < |a| < 1$, it's flatter. A negative $a$ reflects the graph across the $x$-axis.
• Period change ($b$): $y = \tan(bx)$ changes the period to $\frac{\pi}{|b|}$. When $|b| > 1$, the graph compresses horizontally (shorter period). When $0 < |b| < 1$, it stretches horizontally (longer period). The asymptotes shift accordingly.
• Horizontal shift ($h$): $y = \tan(x - h)$ shifts the entire graph right by $h$ units (left if $h$ is negative). The asymptotes and intercepts all shift by the same amount.
• Vertical shift ($k$): $y = \tan x + k$ moves the graph up by $k$ units (down if $k$ is negative).

For example, $y = 2\tan(3x)$ has a period of $\frac{\pi}{3}$ and is vertically stretched by a factor of 2. Its asymptotes occur at $x = \frac{\pi}{6} + \frac{n\pi}{3}$.

Characteristics of the Cotangent Graph

Cotangent is defined as $\cot x = \frac{\cos x}{\sin x}$. It's undefined wherever $\sin x = 0$, so the asymptotes and intercepts swap positions compared to tangent.

• Period: $\pi$
• Vertical asymptotes: $x = n\pi$, where $n$ is any integer
• $x$-intercepts: $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer
• Range: All real numbers
• Odd function: $\cot(-x) = -\cot(x)$

The key visual difference from tangent: between consecutive asymptotes, cotangent falls from $+\infty$ to $-\infty$ (decreasing), while tangent rises (increasing).

Transformations of Cotangent Functions

These follow the same rules as tangent transformations, with the general form $y = a\cot(b(x - h)) + k$:

• Vertical stretch/compression ($a$): Stretches by $|a|$; negative $a$ reflects across the $x$-axis
• Period change ($b$): New period is $\frac{\pi}{|b|}$
• Horizontal shift ($h$): Shifts graph and all asymptotes right by $h$
• Vertical shift ($k$): Moves graph up or down by $k$

Secant and Cosecant Graphs

Since $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$, these graphs are closely tied to cosine and sine. A helpful graphing strategy: sketch the corresponding sine or cosine curve first, then build the reciprocal graph from it.

Where sine or cosine equals $\pm 1$, the reciprocal function also equals $\pm 1$ (these become the "turning points" of the U-shaped branches). Where sine or cosine equals 0, the reciprocal function has a vertical asymptote.

Transformations of Secant and Cosecant

The general forms are $y = a\sec(b(x - h)) + k$ and $y = a\csc(b(x - h)) + k$:

• Vertical stretch ($a$): Stretches by $|a|$. The range becomes $(-\infty, -|a|] \cup [|a|, \infty)$ before any vertical shift. Negative $a$ reflects across the $x$-axis.
• Period change ($b$): New period is $\frac{2\pi}{|b|}$ (note: $2\pi$, not $\pi$, since the base period is $2\pi$).
• Horizontal shift ($h$): Shifts the graph and all asymptotes right by $h$.
• Vertical shift ($k$): Moves the entire graph up or down, shifting the range to $(-\infty, -|a| + k] \cup [|a| + k, \infty)$.

Continuity and Asymptotic Behavior

All four of these functions are continuous on their domains. The discontinuities (vertical asymptotes) occur only at specific, predictable points where the denominator of the underlying ratio equals zero.

When graphing any of these functions, always find the asymptotes first. They form the skeleton of the graph. Then locate the intercepts and a few key points between asymptotes, and the shape will follow naturally.