Graphing Tangent, Cotangent, Secant, and Cosecant

Trigonometry · Axiom Academy

LESSON Graphing Tangent, Cotangent, Secant, and Cosecant Understanding the graphs of all four trigonometric functions with special attention to asymptotes and reciprocal relationships Since tangent equals sine divided by cosine, it becomes undefined whenever cosine equals zero. This happens at x = π/2, 3π/2, 5π/2, etc.—or more generally, at x = π/2 + nπ where n is any integer. Cotangent is undefined wherever sine equals zero, which occurs at x = 0, π, 2π, etc.—or more generally, at x = nπ where n is any integer. Notice how this is "shifted" compared to tangent's asymptotes. Since secant is 1 divided by cosine, it's undefined whenever cosine equals zero—the same places where tangent has asymptotes. The graph of secant consists of U-shaped and ∩-shaped curves that "hug" the cosine graph where |cos(x)| is large, but shoot off to infinity where cos(x) approaches zero. Cosecant is undefined wherever sine equals zero—the same places where cotangent has asymptotes. Like secant, cosecant's graph consists of U-shaped and ∩-shaped curves, but they're shifted by π/2 compared to secant's curves. 5. Comparing All Four Functions Let's see how these four functions compare side by side, with special attention to their asymptote locations and periods:

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