Graphing Tangent and Reciprocal Functions
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn how to graph the tangent function and the reciprocal trig functions (cotangent, secant, cosecant), including their asymptotes, periods, and behavior.
The Concept
Section titled “The Concept”Tangent: y = tan(x)
Section titled “Tangent: y = tan(x)”Tangent is sin(x) / cos(x). Wherever cosine is zero, tangent is undefined, which creates vertical asymptotes.
Key features:
- Period = π (shorter than sine and cosine)
- Vertical asymptotes at x = π/2 + kπ (where cos = 0)
- Passes through the origin (0, 0) and through (π, 0)
- Range: all real numbers (it goes to +∞ and -∞)
- The graph has a distinctive “S” shape between each pair of asymptotes
The red dashed lines in the graph are the asymptotes. The curve approaches them but never touches them.
Reciprocal Functions
Section titled “Reciprocal Functions”These are defined as the reciprocals of the three main trig functions:
- Cotangent: cot(x) = cos(x) / sin(x) = 1 / tan(x)
- Period = π
- Asymptotes at x = kπ (where sin = 0)
- Secant: sec(x) = 1 / cos(x)
- Period = 2π
- Asymptotes where cos = 0 (same places as tangent)
- Cosecant: csc(x) = 1 / sin(x)
- Period = 2π
- Asymptotes where sin = 0
The key idea: wherever the original function (sin or cos) equals zero, the reciprocal function has a vertical asymptote. Wherever the original function equals 1 or -1, the reciprocal function also equals 1 or -1.
Worked Example
Section titled “Worked Example”1. Graph y = tan(x) over one period (-π/2 to π/2)
Key points: passes through (-π/4, -1), (0, 0), (π/4, 1). Approaches +∞ as x approaches π/2 from the left, and -∞ as x approaches -π/2 from the right.
2. Identify asymptotes for y = sec(x)
sec(x) = 1/cos(x), so asymptotes occur where cos(x) = 0:
That’s x = ±π/2, ±3π/2, ±5π/2, and so on.
3. Compare periods
| Function | Period |
|---|---|
| tan(x) | π |
| cot(x) | π |
| sec(x) | 2π |
| csc(x) | 2π |
Tangent and cotangent repeat twice as fast as secant and cosecant.
Real-World Application
Section titled “Real-World Application”These functions appear in:
- Physics (modeling wave interference and resonance)
- Engineering (signal processing and electrical circuits)
- Optics (light wave behavior near boundaries)
- Architecture (structural vibration analysis)
- Navigation (calculating bearings and slopes)
Example: the tangent function models the steepness of a curve at any point. In calculus, the derivative of many functions involves tangent. In engineering, tangent describes the relationship between voltage and current phase in AC circuits.