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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.
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Tan, Cot, Sec, Csc — Domain Holes
What is it?
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.
In simple words
- tan theta = sin/cos, undefined when cos=0, i.e.
- theta = pi/2 + k pi.
- cot undefined when sin=0.
- tan's range is all reals. sec's range is not. Do not mix them.
Important terms
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 = 12 + 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.
What you should learn
- 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.
Examples
- cos(inside)=0 for tan,sec
- sin(inside)=0 for cot,csc
- Solve for x
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.
RANGE
tan, cot: R. sec, csc: |y| >= 1. a sec(cx): |y| >= |a|. Do not say sec ranges over all reals.
Recommended for Class 11
Full school-level explanation
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.
Where it breaks, then range:
WHAT — 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.
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
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.
Remember
- set inside = a hole of the parent
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.
Remember
- |sec|>=1
Examples
- tan: all R
- sec: skip (-1,1)
- csc same skip
How to answer a tan, cot, sec, csc — domain holes question
How it works
- 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.
Remember
- tan undefined at odd multiples of pi/2
- sec, csc miss (-1,1)
- x != k pi/4
- tan all R, sec |y|>=1
- set inside = a hole of the parent
- |sec|>=1
- tan undefined at odd multiples of pi/2
- sec, csc miss (-1,1)
How the ideas connect
Worked: domain of y = csc 4x. sin 4x != 0 4x != k pi x != k pi / 4, k integer
Deeper understanding
Go beyond memorising Tan, Cot, Sec, Csc — Domain Holes. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: domain of y = csc 4x. sin 4x != 0 4x != k pi x != k pi / 4, k integer
Common mistakes
Exam focus
Core Idea
1. HOLES
2. RANGE
Worked Example
Exam focus
Quick revision
- tan undefined at odd multiples of pi/2
- sec, csc miss (-1,1)
- x != k pi/4
- tan all R, sec |y|>=1
- 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
- 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 !=
tan's range is all reals. sec's range is not. Do not mix them.
For long answers on Tan, Cot, Sec, Csc — Domain Holes: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- tan undefined at odd multiples of pi/2
- sec, csc miss (-1,1)
- x != k pi/4
- tan all R, sec |y|>=1
Important formulas
- tan undefined at odd multiples of pi/2
- sec, csc miss (-1,1)
- x != k pi/4
Common exam traps
- tan's range is all reals. sec's range is not. Do not mix them.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
tan's range is all reals. sec's range is not. Do not mix them.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
tan's range is all reals. sec's range is not. Do not mix them.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
tan undefined at odd multiples of pi/2
sec, csc miss (-1,1)
x != k pi/4
tan all R, sec |y|>=1
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
x != k pi/4
- 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
- Formula
- x != k pi/4
- Example
- cos(inside)=0 for tan,sec
- Critical warning
- tan's range is all reals. sec's range is not. Do not mix them.
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Exam Practice
Define
Q1. Define Tan, Cot, Sec, Csc — Domain Holes.
Explain
Q2. Explain: HOLES.
Example
Q3. Give one clear example of Tan, Cot, Sec, Csc — Domain Holes and say why it fits.
Mistake
Q4. State one common mistake students make about Tan, Cot, Sec, Csc — Domain Holes.
Exam note
Q5. Write a short exam note on Tan, Cot, Sec, Csc — Domain Holes: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.