MathematicsChapter 8: Fundamentals of Trigonometry
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) =
Learning objectives
- 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.
Statement
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.
Mathematical form
A=B in the fundamental law. cos(A-A)
A = 1 matches
Where / symbols
- 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.
Relationship
- cos(A-B)=cos A cos B + sin A sin B
Overview
The engine, then a warning:
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.
Examples Unit circle points · Expand squares · Quote after proof
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.
Examples cos cos + sin sin · Write the full right-hand side · Check A=B: cos0=1
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
How to answer a angles and the fundamental law 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
- Worked check: A=B in the fundamental law.
- cos(A-A)=cos 0=1
- right: cos2 A + sin2 A = 1
- matches
In simple words
cos(A-B) is a product mix, not a subtraction of cosines.
Key points
- cos(A-B)=cos A cos B + sin A sin B
- not cos A - cos B
- distance -> cos(A-B)
- mixed products
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Angles and the Fundamental Law
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
Worked Example
- Worked check: A=B in the fundamental law.
- cos(A-A)=cos 0=1
- right: cos2 A + sin2 A = 1
- matches
Common Misconception
- cos(A-B) is a product mix, not a subtraction of cosines.
FBISE Exam Check
- State the rule for angles and the fundamental law before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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