MathematicsChapter 9: Trigonometric Functions

Tan, Cot, Sec, Csc — Domain Holes

tan theta = sin/cos, undefined when cos=0, i.e.

Learning objectives

State the main ideas and notation used in tan, cot, sec, csc — domain holes.
Apply the rules of tan, cot, sec, csc — domain holes in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

tan theta = sin/cos, undefined when cos=0, i.e. theta = pi/2 + k pi. cot undefined when sin=0. sec=1/cos, same holes as tan. csc=1/sin, same holes as cot. Range of tan and cot: all reals. Range of sec, csc: (-inf,-1] U [1,inf). Meaning: the reciprocal functions are large when sin or cos is near 0, and they skip (-1,1) except they never sit in there.

Mathematical form

theta = sin/cos

cos=0

theta = pi/2 + k pi. cot undefined when sin

sec=1/cos

Where / symbols

TAN, COT, SEC, CSC — DOMAIN HOLES: tan theta = sin/cos, undefined when cos=0, i.e. theta = pi/2 + k pi. cot undefined when sin=0. sec=1/cos, same holes as tan. csc=1/sin, same holes as cot. Range of tan and cot: all reals. Range of sec, csc: (-inf,-1] U [1,inf). Meaning: the reciprocal functions are large when sin or cos is near 0, and they skip (-1,1) except they never sit in there.
1. HOLES: y = tan x: x != pi/2 + k pi. y = 4 tan(pi x): set pi x = pi/2 + k pi, x = 1/2 + k. y = csc 4x: 4x != k pi, x != k pi/4.
2. RANGE: tan, cot: R. sec, csc: |y| >= 1. a sec(cx): |y| >= |a|. Do not say sec ranges over all reals.

Overview

Where it breaks, then range:

Holes

y = tan x: x != pi/2 + k pi. y = 4 tan(pi x): set pi x = pi/2 + k pi, x = 12 + k. y = csc 4x: 4x != k pi, x != k pi/4.

Examples cos(inside)=0 for tan,sec · sin(inside)=0 for cot,csc · Solve for x

Range

tan, cot: R. sec, csc: |y| >= 1. a sec(cx): |y| >= |a|. Do not say sec ranges over all reals.

Examples tan: all R · sec: skip (-1,1) · csc same skip

WHAT — TAN, COT, SEC, CSC — DOMAIN HOLES

RULE — 1. HOLES

y = tan x: x != pi/2 + k pi. y = 4 tan(pi x): set pi x = pi/2 + k pi, x = 12 + k. y = csc 4x: 4x != k pi, x != k pi/4.

METHOD — 2. RANGE

tan, cot: R. sec, csc: |y| >= 1. a sec(cx): |y| >= |a|. Do not say sec ranges over all reals.

WORKED EXAMPLE

Worked: domain of y = csc 4x. sin 4x != 0 4x != k pi x != k pi / 4, k integer

How to answer a tan, cot, sec, csc — domain holes question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

domain of y = csc 4x.
sin 4x != 0
4x != k pi
x != k pi / 4, k integer

In simple words

tan's range is all reals. sec's range is not. Do not mix them.

Key points

tan undefined at odd multiples of pi/2
sec, csc miss (-1,1)
set inside = a hole of the parent
|sec|>=1

Exam preparation

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