Tangent and cotangent functions are the wild cards of trigonometry. They're like rollercoasters, zooming up to infinity and plummeting back down. These functions have no bounds, stretching endlessly in both directions, making them unique among trig functions.

Unlike their well-behaved cousins sine and cosine, tangent and cotangent have gaps in their graphs called asymptotes. These create a pattern of repeating curves that loop infinitely. Understanding their quirks is key to mastering advanced trig concepts.

Graphs of Tangent and Cotangent Functions

Tangent function key features

• General form $y = A \tan(B(x - C)) + D$ where A amplifies, B affects period, C shifts horizontally, D shifts vertically
• Period $\frac{\pi}{|B|}$ determines frequency of repetition (smaller B, longer period)
• Amplitude undefined extends infinitely both directions
• Vertical asymptotes at $x = \frac{\pi}{2} + n\pi$ (n is integer) create discontinuities
• Parent function $y = \tan(x)$ has period $\pi$, domain all reals except $\frac{\pi}{2} + n\pi$, range all reals
• Key points: x-intercepts at multiples of $\pi$, y-intercept at (0, 0)

Cotangent function key features

• General form $y = A \cot(B(x - C)) + D$ mirrors tangent structure
• Period $\frac{\pi}{|B|}$ similar to tangent but shifted
• Amplitude undefined extends infinitely both directions
• Vertical asymptotes at $x = n\pi$ (n is integer) create discontinuities
• Parent function $y = \cot(x)$ has period $π$, domain all reals except $n\pi$, range all reals
• Key points: x-intercepts at odd multiples of $\frac{\pi}{2}$, no y-intercept (asymptote at x = 0)

Tangent vs cotangent relationship

• Reciprocal functions $\cot(x) = \frac{1}{\tan(x)}$ inverse relationship
• Complementary angles $\tan(x) = \cot(\frac{\pi}{2} - x)$ connect functions
• Graphs reflect over line $y = x$ mirror image property
• Tangent's vertical asymptotes are cotangent's x-intercepts and vice versa interchangeable features

Domain and range of trigonometric functions

• Tangent: domain all reals except $\frac{\pi}{2} + n\pi$, range all reals
• Cotangent: domain all reals except $n\pi$, range all reals
• Transformations affect domain and range:
  • Vertical stretch/compression: no change to domain or range
  • Horizontal stretch/compression: alters asymptote spacing
  • Vertical shift: range shifts, domain unchanged
  • Horizontal shift: asymptotes and domain restrictions move