MathematicsChapter 9: Trigonometric Functions
Even, Odd, and Periodicity
Even: f(-x)=f(x) (cos, sec).
Learning objectives
- 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.
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. 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'.
Mathematical form
T = 2 pi / |23| = 2 pi * 32 = 3 pi
y = sin(2x/3). parent period 2 pi
c = 23 T
Where / symbols
- 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|.
Overview
Mirror, then the repeat length:
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.
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|.
Examples sin, cos: 2 pi/|c| · tan: pi/|c| · Phase d does not change T
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
How to answer a even, odd, and periodicity 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
- period of y = sin(2x/3).
- parent period 2 pi, c = 23
- T = 2 pi / |23| = 2 pi * 32 = 3 pi
- not 2 pi
In simple words
tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
Key points
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
- f(-x) vs f(x)
- divide by |c|
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Even, Odd, and Periodicity
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
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.
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
Common Misconception
- tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
FBISE Exam Check
- State the rule for even, odd, and periodicity before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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|.
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