8.2 Graphs of the Other Trigonometric Functions

Trigonometric functions beyond sine and cosine offer unique patterns and behaviors. Tangent, secant, cosecant, and cotangent each have distinct graphs with their own periods, asymptotes, and symmetries.

These functions can be transformed through shifts, stretches, and reflections. Understanding their graphs and transformations helps in modeling periodic phenomena and solving complex trigonometric equations.

Graphs of Tangent, Secant, Cosecant, and Cotangent Functions

Graph of tangent function

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• Periodic function repeats its pattern every $\pi$ units
• Vertical asymptotes occur at $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer
  • Graph approaches positive or negative infinity as $x$ approaches these values from either side
• $x$-intercepts located at $x = n\pi$, where $n$ is any integer
• Graph passes through the origin $(0, 0)$
• Odd function $\tan(-x) = -\tan(x)$ exhibits symmetry about the origin
• Domain is all real numbers except where vertical asymptotes occur

Variations of tangent functions

• Vertical shifts $y = \tan x + k$ move the graph up or down by $k$ units
  • Positive $k$ values shift the graph upward
  • Negative $k$ values shift the graph downward
• Horizontal shifts $y = \tan(x - h)$ move the graph left or right by $h$ units
  • Positive $h$ values shift the graph to the right
  • Negative $h$ values shift the graph to the left
• Amplitude changes $y = a \tan x$ stretch or compress the graph vertically
  • $|a| > 1$ vertically stretches the graph by a factor of $|a|$
  • $0 < |a| < 1$ vertically compresses the graph by a factor of $|a|$
  • Negative $a$ values reflect the graph across the $x$-axis
• Period changes $y = \tan(bx)$ alter the repetition interval of the function
  • Period becomes $\frac{\pi}{|b|}$
  • $|b| > 1$ horizontally compresses the graph
  • $0 < |b| < 1$ horizontally stretches the graph

Secant vs cosecant graphs

• Both functions have a period of $2\pi$ and lack $x$-intercepts
• Vertical asymptotes differ in location
  • $y = \sec x$ has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer
  • $y = \csc x$ has vertical asymptotes at $x = n\pi$, where $n$ is any integer
• $y = \sec x$ is an even function symmetric about the $y$-axis
• $y = \csc x$ is an odd function symmetric about the origin
• Both are reciprocal functions of cosine and sine, respectively

Transformations of secant and cosecant

• Vertical shifts $y = \sec x + k$ and $y = \csc x + k$ move the graph up or down by $k$ units
• Horizontal shifts $y = \sec(x - h)$ and $y = \csc(x - h)$ move the graph left or right by $h$ units
• Amplitude changes $y = a \sec x$ and $y = a \csc x$ stretch or compress the graph vertically by a factor of $|a|$
  • Negative $a$ values reflect the graph across the $x$-axis
• Period changes $y = \sec(bx)$ and $y = \csc(bx)$ alter the repetition interval to $\frac{2\pi}{|b|}$

Characteristics of cotangent graph

• Periodic function with a period of $\pi$
• Vertical asymptotes occur at $x = n\pi$, where $n$ is any integer
• $x$-intercepts located at $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer
• Odd function $\cot(-x) = -\cot(x)$ exhibits symmetry about the origin
• Range of cotangent function is all real numbers

Transformations of cotangent functions

• Vertical shifts $y = \cot x + k$ move the graph up or down by $k$ units
• Horizontal shifts $y = \cot(x - h)$ move the graph left or right by $h$ units
• Amplitude changes $y = a \cot x$ stretch or compress the graph vertically by a factor of $|a|$
  • Negative $a$ values reflect the graph across the $x$-axis
• Period changes $y = \cot(bx)$ alter the repetition interval to $\frac{\pi}{|b|}$

Additional Concepts

• Continuity and discontinuity: Trigonometric functions are continuous except at points of discontinuity (vertical asymptotes)
• Function composition: Combining trigonometric functions with other functions can create new, complex relationships
• Inverse trigonometric functions: Arcsine, arccosine, and arctangent are examples that "undo" their corresponding trigonometric functions