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Not Linear
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
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Not Linear
What is it?
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
In simple words
- cos(A-B) != cos A - cos B.
- cos cos + sin sin
- Write the full right-hand side
- State the rule, show the working, and check the answer.
Important terms
NOT LINEAR: cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
NOT LINEAR: cos(A-B) != cos A - cos B.
What you should learn
- State the rule or meaning of not linear in easy words.
- Apply not linear in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- cos cos + sin sin
- Write the full right-hand side
- Check A=B: cos0=1
MAIN IDEA
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
EXAM USE
Write the rule first: cos(A-B) != cos A - cos B. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
Understand the idea, then apply it:
WHAT — NOT LINEAR
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
WHY — Why it matters
This rule is a separate learning unit inside angles and the fundamental law. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: cos(A-B) != cos A - cos B. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
Remember
- mixed products
Examples
- cos cos + sin sin
- Write the full right-hand side
- Check A=B: cos0=1
EXAM USE
Write the rule first: cos(A-B) != cos A - cos B. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a not linear question
How it works
- 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.
Remember
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- not cos A - cos B
- mixed products
- focused answer
- accurate support
- mixed products
Deeper understanding
Go beyond memorising Not Linear. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- NOT LINEAR: cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixe
- NOT LINEAR: cos(A-B) != cos A - cos B.
- WHAT — NOT LINEAR: cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixe
cos(A-B) is a product mix, not a subtraction of cosines.
For long answers on Not Linear: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- NOT LINEAR: cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixe
Important formulas
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
Common exam traps
- cos(A-B) is a product mix, not a subtraction of cosines.
- Do not choose a formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
cos(A-B) is a product mix, not a subtraction of cosines.
- Mistake
Do not choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- Mistake
cos(A-B) is a product mix, not a subtraction of cosines.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
mixed products
cos(A-B)=cos A cos B + sin A sin B
not cos A - cos B
NOT LINEAR: cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixe
cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
cos(A-B)=cos A cos B + sin A sin B
- Definition
- cos(A-B) != cos A - cos B. sin(A+B) != sin A + sin B. Those are the most common slips. Keep the mixed products.
- Formula
- cos(A-B)=cos A cos B + sin A sin B
- Example
- cos cos + sin sin
- Critical warning
- cos(A-B) is a product mix, not a subtraction of cosines.
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Exam Practice
Define
Q1. Define Not Linear.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of Not Linear and say why it fits.
Mistake
Q4. State one common mistake students make about Not Linear.
Exam note
Q5. Write a short exam note on Not Linear: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.