8.2 Graphs of the Other Trigonometric Functions

Tangent, secant, cosecant, and cotangent functions expand on the basic sine and cosine. These trig functions have unique patterns and properties, like periods, ranges, and asymptotes, that set them apart from their simpler cousins.

Understanding how to graph and transform these functions is key. You'll learn about their domains, ranges, and special features like vertical asymptotes. This knowledge helps solve more complex trig problems and model real-world periodic phenomena.

Graphing Tangent, Secant, Cosecant, and Cotangent Functions

Graph of tangent function

• Periodic function repeats its pattern every $\pi$ units
• Vertical asymptotes occur at $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer, when the cosine function equals 0
• Graph passes through the origin (0, 0)
• As $x$ approaches vertical asymptotes from the left, $\tan x$ approaches positive or negative infinity (depending on the quadrant)
• As $x$ approaches vertical asymptotes from the right, $\tan x$ approaches negative or positive infinity (opposite of left-hand limit)
• Can take on any real value between negative and positive infinity
• Tangent is an odd function, exhibiting symmetry about the origin

Transformations of tangent functions

• Vertical shifts $y = \tan x + k$ move the graph up by $k$ units if $k > 0$, down by $|k|$ units if $k < 0$
• Horizontal shifts $y = \tan(x - h)$ move the graph right by $h$ units if $h > 0$, left by $|h|$ units if $h < 0$
• Vertical stretches/compressions $y = a \tan x$ stretch the graph vertically by a factor of $|a|$ if $|a| > 1$, compress it if $0 < |a| < 1$ (reflection across $x$-axis if $a < 0$)
• Horizontal stretches/compressions $y = \tan(bx)$ compress the graph horizontally by a factor of $|b|$ if $|b| > 1$, stretch it if $0 < |b| < 1$ (period becomes $\frac{\pi}{|b|}$)
• The amplitude of tangent function is undefined due to its vertical asymptotes

Secant vs cosecant graphs

• Secant $y = \sec x$ has a period of $2\pi$, cosecant $y = \csc x$ also has a period of $2\pi$
• Both have vertical asymptotes
  1. Secant: at $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer (cosine equals 0)
  2. Cosecant: at $x = n\pi$ where $n$ is any integer (sine equals 0)
• Secant has a minimum of 1 and maximum of -1, cosecant has no max/min values
• Secant range: $(-\infty, -1] \cup [1, \infty)$, cosecant range: $(-\infty, -1] \cup [1, \infty)$
• Both undefined at their respective vertical asymptotes
• Secant is an even function, showing symmetry about the y-axis

Cotangent function analysis

• Reciprocal of tangent function $y = \cot x = \frac{1}{\tan x}$
• Period of $\pi$, same as tangent function
• Vertical asymptotes at $x = n\pi$ where $n$ is any integer (sine equals 0)
• As $x$ approaches vertical asymptotes from left, $\cot x$ approaches negative or positive infinity (depending on quadrant)
• As $x$ approaches vertical asymptotes from right, $\cot x$ approaches positive or negative infinity (opposite of left-hand limit)
• Range includes all real numbers from negative to positive infinity

Transformations of reciprocal functions

• Secant, cosecant, cotangent transformations follow same rules as tangent
  • Vertical shifts: $y = \sec x + k$, $y = \csc x + k$, $y = \cot x + k$
  • Horizontal shifts: $y = \sec(x - h)$, $y = \csc(x - h)$, $y = \cot(x - h)$
  • Vertical stretches/compressions: $y = a \sec x$, $y = a \csc x$, $y = a \cot x$
  • Horizontal stretches/compressions: $y = \sec(bx)$, $y = \csc(bx)$, $y = \cot(bx)$
• Transformed secant and cosecant period becomes $\frac{2\pi}{|b|}$, cotangent period becomes $\frac{\pi}{|b|}$

Properties of trigonometric functions

• Tangent:
  • Domain: all real numbers except $x \neq \frac{\pi}{2} + n\pi$ where $n$ is any integer
  • Range: $(-\infty, \infty)$
  • Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer
• Secant:
  • Domain: all real numbers except $x \neq \frac{\pi}{2} + n\pi$ where $n$ is any integer
  • Range: $(-\infty, -1] \cup [1, \infty)$
  • Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer
• Cosecant:
  • Domain: all real numbers except $x \neq n\pi$ where $n$ is any integer
  • Range: $(-\infty, -1] \cup [1, \infty)$
  • Vertical asymptotes: $x = n\pi$ where $n$ is any integer
• Cotangent:
  • Domain: all real numbers except $x \neq n\pi$ where $n$ is any integer
  • Range: $(-\infty, \infty)$
  • Vertical asymptotes: $x = n\pi$ where $n$ is any integer

Continuity of trigonometric functions

• Tangent, secant, cosecant, and cotangent functions are continuous on their domains
• Discontinuities occur at vertical asymptotes where these functions are undefined