Graphs of Tangent, Cotangent, Secant & Cosecant – Key Vocabulary

A set of vocabulary flashcards covering the key terms, identities, and transformation parameters involved in graphing the tangent, cotangent, secant, and cosecant functions as discussed in Module 5.

Secant Function (sec x)

The reciprocal of cosine; sec x = 1 ⁄ cos x. Its period is 2π and it has vertical asymptotes where cos x = 0 (odd multiples of π⁄2).

Cosecant Function (csc x)

The reciprocal of sine; csc x = 1 ⁄ sin x. Its period is 2π and it has vertical asymptotes where sin x = 0 (integer multiples of π).

Tangent Function (tan x)

Defined by tan x = sin x ⁄ cos x. It repeats every π and has vertical asymptotes where cos x = 0 (odd multiples of π⁄2).

Cotangent Function (cot x)

Defined by cot x = cos x ⁄ sin x. It repeats every π and has vertical asymptotes where sin x = 0 (integer multiples of π).

Reciprocal Identity

Any identity expressing a trig function as the reciprocal of another, e.g., sec x = 1⁄cos x and csc x = 1⁄sin x.

Quotient Identity

Identities that express tangent and cotangent as quotients: tan x = sin x⁄cos x and cot x = cos x⁄sin x.

Period

The smallest positive number p for which a periodic function satisfies f(x) = f(x + p). Sin, cos, sec, csc have period 2π; tan and cot have period π.

Domain of sec x

All real x except where cos x = 0; i.e., x ≠ (2n + 1)π⁄2 for any integer n.

Domain of csc x

All real x except where sin x = 0; i.e., x ≠ nπ for any integer n.

Vertical Asymptote

A vertical line x = a that a graph approaches but never crosses because the function grows without bound as x approaches a.

Vertical Stretch / Compression

A transformation produced by multiplying a function by a factor |a|. |a| > 1 stretches the graph; 0 < |a| < 1 compresses it.

Horizontal Stretch / Compression

A transformation caused by a coefficient b inside the function, f(bx). The new period equals original period⁄b; |b| > 1 compresses, 0 < |b| < 1 stretches.

Translation (Shift)

Moving a graph without changing its shape: horizontally by c (phase shift) and vertically by d (vertical shift).

Phase Shift

The horizontal displacement of a periodic graph determined by the value of c in f(x − c).

Vertical Shift

The upward or downward displacement of a graph determined by adding d to the function value.

Transformation

Any combination of stretches, reflections, and translations applied to a parent graph to obtain a new graph.

Parent Function

The simplest form of a function (e.g., y = csc x) that serves as the basis for transformations.

Local Maximum

A highest point within a small neighborhood of the graph (a “peak”) between two vertical asymptotes for csc or sec.

Local Minimum

A lowest point within a small neighborhood of the graph (a “valley”) between two vertical asymptotes for csc or sec.

Unit Circle

A circle of radius 1 centered at the origin used to define trigonometric functions via coordinates of points on the circle.

Reference Angle

The acute angle formed between the terminal side of an angle and the x-axis; useful for evaluating trig functions.

Coefficient a

The constant multiplying the function (outside); controls vertical stretch, compression, and reflection over the x-axis.

Coefficient b

The constant multiplying x (inside); controls horizontal stretch/compression and determines new period (period = parent ⁄ |b|).

Coefficient c

The horizontal translation parameter in f(x − c); positive c shifts the graph right, negative left.

Coefficient d

The vertical translation parameter; positive d shifts the graph up, negative down.

Vertical Distance Between Valleys and Peaks

For csc or sec, the distance between a local minimum and the adjacent local maximum within one loop; equals 2|a| in the standard form a csc x + d or a sec x + d.

Asymptote Spacing for csc/sec

Vertical asymptotes occur at integer multiples of π for csc and at odd multiples of π⁄2 for sec, repeating every π.

Asymptote Spacing for tan/cot

Vertical asymptotes of tan occur at x = (2n + 1)π⁄2; for cot at x = nπ, with spacing π between successive asymptotes.

Continuity of Reciprocal Functions

Sec x and csc x are continuous wherever cos x ≠ 0 or sin x ≠ 0, respectively; discontinuities occur only at their asymptotes.

Period Copy-and-Paste

Technique of graphing one full period of a periodic function and then repeating that segment along the x-axis to extend the graph.