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Even, Odd, and Periodicity
Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi. Of tan, cot: pi. Of sin(cx): 2 pi/|c|. Of tan(cx): pi/|c|. Meaning: even/odd is a mirror test; period is 'how soon the wave repeats'.
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Even, Odd, and Periodicity
What is it?
Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi. Of tan, cot: pi. Of sin(cx): 2 pi/|c|. Of tan(cx): pi/|c|. Meaning: even/odd is a mirror test; period is 'how soon the wave repeats'.
In simple words
- Even: f(-x)=f(x) (cos, sec).
- Odd: f(-x)=-f(x) (sin, tan, cot, csc).
- A period T>0 means f(x+T)=f(x) for all x in the domain.
- tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
Important terms
EVEN, ODD, AND PERIODICITY: Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi. Of tan, cot: pi. Of sin(cx): 2 pi/|c|. Of tan(cx): pi/|c|. Meaning: even/odd is a mirror test; period is 'how soon the wave repeats'.
1. EVEN / ODD: cos(-x)=cos x. sin(-x)=-sin x. A sum of even is even. sin is odd so a graph through the origin, 180 rotational symmetry. Neither: e.g. sin x + cos x is neither.
2. PERIOD: Smallest positive T. sin(2x) has period pi, not 2 pi. tan(2x) has period pi/2. Adding a constant does not change period. a+b sin(cx+d): still 2 pi/|c|.
What you should learn
- State the main ideas and notation used in even, odd, and periodicity.
- Apply the rules of even, odd, and periodicity in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- cos even
- tan odd
- Neither: mix
EVEN / ODD
cos(-x)=cos x. sin(-x)=-sin x. A sum of even is even. sin is odd so a graph through the origin, 180 rotational symmetry. Neither: e.g. sin x + cos x is neither.
PERIOD
Smallest positive T. sin(2x) has period pi, not 2 pi. tan(2x) has period pi/2. Adding a constant does not change period. a+b sin(cx+d): still 2 pi/|c|.
Recommended for Class 11
Full school-level explanation
Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi. Of tan, cot: pi. Of sin(cx): 2 pi/|c|. Of tan(cx): pi/|c|. Meaning: even/odd is a mirror test; period is 'how soon the wave repeats'.
Mirror, then the repeat length:
WHAT — EVEN, ODD, AND PERIODICITY
Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi. Of tan, cot: pi. Of sin(cx): 2 pi/|c|. Of tan(cx): pi/|c|. Meaning: even/odd is a mirror test; period is 'how soon the wave repeats'.
RULE — 1. EVEN / ODD
cos(-x)=cos x. sin(-x)=-sin x. A sum of even is even. sin is odd so a graph through the origin, 180 rotational symmetry. Neither: e.g. sin x + cos x is neither.
METHOD — 2. PERIOD
Smallest positive T. sin(2x) has period pi, not 2 pi. tan(2x) has period pi/2. Adding a constant does not change period. a+b sin(cx+d): still 2 pi/|c|.
WORKED EXAMPLE
Worked: period of y = sin(2x/3). parent period 2 pi, c = 23 T = 2 pi / |23| = 2 pi * 32 = 3 pi not 2 pi
EVEN / ODD
cos(-x)=cos x. sin(-x)=-sin x. A sum of even is even. sin is odd so a graph through the origin, 180 rotational symmetry. Neither: e.g. sin x + cos x is neither.
Remember
- f(-x) vs f(x)
Examples
- cos even
- tan odd
- Neither: mix
PERIOD
Smallest positive T. sin(2x) has period pi, not 2 pi. tan(2x) has period pi/2. Adding a constant does not change period. a+b sin(cx+d): still 2 pi/|c|.
Remember
- divide by |c|
Examples
- sin, cos: 2 pi/|c|
- tan: pi/|c|
- Phase d does not change T
How to answer a even, odd, and periodicity 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
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
- 3 pi
- tan period pi
- f(-x) vs f(x)
- divide by |c|
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
How the ideas connect
Worked: period of y = sin(2x/3). parent period 2 pi, c = 23 T = 2 pi / |23| = 2 pi * 32 = 3 pi not 2 pi
Deeper understanding
Go beyond memorising Even, Odd, and Periodicity. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: period of y = sin(2x/3). parent period 2 pi, c = 23 T = 2 pi / |23| = 2 pi * 32 = 3 pi not 2 pi
Common mistakes
Exam focus
Core Idea
1. EVEN / ODD
2. PERIOD
Worked Example
Exam focus
Quick revision
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
- 3 pi
- tan period pi
- EVEN, ODD, AND PERIODICITY: Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) f
- 1. EVEN / ODD: cos(-x)=cos x. sin(-x)=-sin x. A sum of even is even. sin is odd so a graph through the origin, 180
tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
For long answers on Even, Odd, and Periodicity: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
- 3 pi
- tan period pi
Important formulas
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
- 3 pi
Common exam traps
- tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
- 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 repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
- 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 repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sin odd, cos even
period of sin(cx) is 2 pi/|c|
3 pi
tan period pi
Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi
- Definition
- Even: f(-x)=f(x) (cos, sec). Odd: f(-x)=-f(x) (sin, tan, cot, csc). A period T>0 means f(x+T)=f(x) for all x in the domain. Fundamental period of sin, cos: 2 pi
- Example
- cos even
- Critical warning
- tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
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Exam Practice
Define
Q1. Define Even, Odd, and Periodicity.
Explain
Q2. Explain: EVEN / ODD.
Example
Q3. Give one clear example of Even, Odd, and Periodicity and say why it fits.
Mistake
Q4. State one common mistake students make about Even, Odd, and Periodicity.
Exam note
Q5. Write a short exam note on Even, Odd, and Periodicity: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.