MathematicsChapter 9: Trigonometric Functions
Range of Sine and Cosine
-1 <= sin(anything) <= 1.
Learning objectives
- State the rule or meaning of range of sine and cosine in easy words.
- Apply range of sine and cosine in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Where / symbols
- RANGE OF SINE AND COSINE
- -1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
- RANGE
- -1 <= sin(anything) <= 1.
Overview
Understand the idea, then apply it:
Main Idea
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Examples [-1,1] · [-7,7] · a+b sin: a plus minus |b|
Exam Use
Write the rule first: -1 <= sin(anything) <= 1. 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 — RANGE OF SINE AND COSINE
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
WHY — Why it matters
This rule is a separate learning unit inside domain and range of sin and cos. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: -1 <= sin(anything) <= 1. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a range of sine and cosine 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
The 4 in cos 4x changes the period, not the range. Range still uses |7|.
Key points
- stretch |b|, shift a
- domain of sin, cos: all real
- range [-1,1]
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Range of Sine and Cosine
Sticky Notes · Quick Revision
Core Idea
- -1 <= sin(anything) <= 1.
What to Know
- -1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Example or Evidence
- [-1,1]
- [-7,7]
- a+b sin: a plus minus |b|
How to Answer
- Write the rule first: -1 <= sin(anything) <= 1. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- The 4 in cos 4x changes the period, not the range. Range still uses |7|.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- -1 <= sin(anything) <= 1.
- [-1,1]
- [-7,7]
- State the rule, show the working, and check the answer.
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