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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
- The fundamental law in this unit is built from the distance formula: two points on the unit circle, expand |P-Q|^2, and read off cos(A-B) = cos A cos B + sin A sin B.
- That one identity is the engine.
- Then you get cos(A+B), sin(A±B), tan(A±B) as deductions.
- cos(A-B) is a product mix, not a subtraction of cosines.
Important terms
ANGLES AND THE FUNDAMENTAL LAW: The fundamental law in this unit is built from the distance formula: two points on the unit circle, expand |P-Q|^2, and read off cos(A-B) = cos A cos B + sin A sin B. That one identity is the engine. Then you get cos(A+B), sin(A±B), tan(A±B) as deductions. Meaning: cosine of a difference is not cos A - cos B. It is a mix of sin and cos.
1. COS OF DIFFERENCE: Distance on the unit circle between angles A and B. Expand (cos A - cos B)^2 + (sin A - sin B)^2 = 2-2 cos(A-B) school. Rearrange to the law. You may quote it once proved.
2. 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.
What you should learn
- State the main ideas and notation used in angles and the fundamental law.
- Apply the rules of angles and the fundamental law in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- cos cos + sin sin
- Write the full right-hand side
- Check A=B: cos0=1
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.
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.
The engine, then a warning:
WHAT — ANGLES AND THE FUNDAMENTAL LAW
The fundamental law in this unit is built from the distance formula: two points on the unit circle, expand |P-Q|^2, and read off cos(A-B) = cos A cos B + sin A sin B. That one identity is the engine. Then you get cos(A+B), sin(A±B), tan(A±B) as deductions. Meaning: cosine of a difference is not cos A - cos B. It is a mix of sin and cos.
RULE — 1. COS OF DIFFERENCE
Distance on the unit circle between angles A and B. Expand (cos A - cos B)^2 + (sin A - sin B)^2 = 2-2 cos(A-B) school. Rearrange to the law. You may quote it once proved.
METHOD — 2. 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.
WORKED EXAMPLE
Worked check: A=B in the fundamental law. cos(A-A)=cos 0=1 right: cos^2 A + sin^2 A = 1 matches
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.
Remember
- mixed products
Examples
- cos cos + sin sin
- Write the full right-hand side
- Check A=B: cos0=1
Remember
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- distance -> cos(A-B)
- mixed products
How the ideas connect
Worked check: A=B in the fundamental law. cos(A-A)=cos 0=1 right: cos^2 A + sin^2 A = 1 matches
Deeper understanding
Go beyond memorising Not Linear. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked check: A=B in the fundamental law. cos(A-A)=cos 0=1 right: cos^2 A + sin^2 A = 1 matches
Common mistakes
Exam focus
Core Idea
1. COS OF DIFFERENCE
2. NOT LINEAR
Worked Example
Exam focus
Quick revision
- mixed products
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- ANGLES AND THE FUNDAMENTAL LAW: The fundamental law in this unit is built from the distance formula: two points on the unit circle,
- 1. COS OF DIFFERENCE: Distance on the unit circle between angles A and B. Expand (cos A - cos B)^2 + (sin A - sin B)^2 = 2
- 2. 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
- ANGLES AND THE FUNDAMENTAL LAW: The fundamental law in this unit is built from the distance formula: two points on the unit circle,
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 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
cos(A-B) is a product mix, not a subtraction of cosines.
- 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
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
ANGLES AND THE FUNDAMENTAL LAW: The fundamental law in this unit is built from the distance formula: two points on the unit circle,
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: NOT LINEAR.
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.