MathematicsChapter 9: Trigonometric Functions
Periods of Trigonometric Functions
Smallest positive T. sin(2x) has period pi, not 2 pi.
Learning objectives
- State the rule or meaning of periods of trigonometric functions in easy words.
- Apply periods of trigonometric functions in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
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|.
Where / symbols
- PERIODS OF TRIGONOMETRIC FUNCTIONS
- 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|.
- PERIOD
- Smallest positive T.
Overview
Understand the idea, then apply it:
Main Idea
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
Exam Use
Write the rule first: Smallest positive T. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Examples State the central idea · Use one accurate detail · Check the final response
WHAT — PERIODS OF TRIGONOMETRIC FUNCTIONS
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|.
WHY — Why it matters
This rule is a separate learning unit inside even, odd, and periodicity. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Smallest positive T. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a periods of trigonometric functions question
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
In simple words
tan repeats every pi, not every 2 pi. sin(cx) uses 2 pi/|c|, tan(cx) uses pi/|c|.
Key points
- divide by |c|
- sin odd, cos even
- period of sin(cx) is 2 pi/|c|
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Periods of Trigonometric Functions
Sticky Notes · Quick Revision
Core Idea
- Smallest positive T.
What to Know
- 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|.
Example or Evidence
- sin, cos: 2 pi/|c|
- tan: pi/|c|
- Phase d does not change T
How to Answer
- Write the rule first: Smallest positive T. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- 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, show the working, and check the answer.
60-second revision
- Smallest positive T.
- sin, cos: 2 pi/|c|
- tan: pi/|c|
- State the rule, show the working, and check the answer.
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