MathematicsChapter 8: Fundamentals of Trigonometry
Product To Sum
2 sin A cos B = sin(A+B)+sin(A-B).
Learning objectives
- State the main ideas and notation used in product to sum.
- Apply the rules of product to sum in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
2 sin A cos B = sin(A+B)+sin(A-B). 2 cos A cos B = cos(A+B)+cos(A-B). 2 sin A sin B = cos(A-B)-cos(A+B). (Signs: the sine-sine one is a difference of cosines.) Meaning: a product of two waves becomes a sum, useful to integrate later and to simplify a product in an identity question now.
Mathematical form
B = sin(A+B)+sin(A-B). 2 cos A cos B
B = cos(A-B)-cos(A+B). (Signs
Where / symbols
- PRODUCT TO SUM
- 2 sin A cos B = sin(A+B)+sin(A-B). 2 cos A cos B = cos(A+B)+cos(A-B). 2 sin A sin B = cos(A-B)-cos(A+B). (Signs: the sine-sine one is a difference of cosines.) Meaning: a product of two waves becomes a sum, useful to integrate later and to simplify a product in an identity question now.
- 1. 2 SIN COS
- From expanding sin(A+B)+sin(A-B). If A=B, 2 sin A cos A = sin 2A, which you already know. Keep the 2 on the left as the book writes it.
- 2. COS COS AND SIN SIN
- 2 cos A cos B = cos(A+B)+cos(A-B). 2 sin A sin B = cos(A-B)-cos(A+B) — minus on the sum cosine. Write the three as a block to memorise.
Relationship
- 2 sin A cos B = sin(A+B)+sin(A-B)
- 2 sin A sin B = cos(A-B)-cos(A+B)
Overview
Sin-cos, then the two cosines:
2 Sin Cos
From expanding sin(A+B)+sin(A-B). If A=B, 2 sin A cos A = sin 2A, which you already know. Keep the 2 on the left as the book writes it.
Examples Prosthaphaeresis named · 2 in front · A=B recovers double
Cos Cos and Sin Sin
2 cos A cos B = cos(A+B)+cos(A-B). 2 sin A sin B = cos(A-B)-cos(A+B) — minus on the sum cosine. Write the three as a block to memorise.
Examples Both plus for cos cos · Minus for sin sin · Check A=B: 2 sin^2 A = 1-cos 2A
WHAT — PRODUCT TO SUM
2 sin A cos B = sin(A+B)+sin(A-B). 2 cos A cos B = cos(A+B)+cos(A-B). 2 sin A sin B = cos(A-B)-cos(A+B). (Signs: the sine-sine one is a difference of cosines.) Meaning: a product of two waves becomes a sum, useful to integrate later and to simplify a product in an identity question now.
RULE — 1. 2 SIN COS
From expanding sin(A+B)+sin(A-B). If A=B, 2 sin A cos A = sin 2A, which you already know. Keep the 2 on the left as the book writes it.
METHOD — 2. COS COS AND SIN SIN
2 cos A cos B = cos(A+B)+cos(A-B). 2 sin A sin B = cos(A-B)-cos(A+B) — minus on the sum cosine. Write the three as a block to memorise.
WORKED EXAMPLE
Worked: 2 sin 75 cos 15. = sin 90 + sin 60 = 1 + sqrt(3)/2
How to answer a product to sum 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
- 2 sin 75 cos 15.
- = sin 90 + sin 60
- = 1 + sqrt(3)/2
In simple words
2 sin A sin B is NOT sin(A+B)+sin(A-B). That one is 2 sin cos.
Key points
- 2 sin A cos B = sin(A+B)+sin(A-B)
- 2 sin A sin B = cos(A-B)-cos(A+B)
- expand compounds
- sin sin: cos minus cos
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Product To Sum
Sticky Notes · Quick Revision
Core Idea
- 2 sin A cos B = sin(A+B)+sin(A-B).
- 2 cos A cos B = cos(A+B)+cos(A-B).
- 2 sin A sin B = cos(A-B)-cos(A+B).
1. 2 SIN COS
- From expanding sin(A+B)+sin(A-B).
- If A=B, 2 sin A cos A = sin 2A, which you already know.
- Keep the 2 on the left as the book writes it.
2. COS COS AND SIN SIN
- 2 cos A cos B = cos(A+B)+cos(A-B).
- 2 sin A sin B = cos(A-B)-cos(A+B) — minus on the sum cosine.
- Write the three as a block to memorise.
Worked Example
- Worked: 2 sin 75 cos 15.
- = sin 90 + sin 60
- = 1 + sqrt(3)/2
Common Misconception
- 2 sin A sin B is NOT sin(A+B)+sin(A-B). That one is 2 sin cos.
FBISE Exam Check
- State the rule for product to sum before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 2 sin A cos B = sin(A+B)+sin(A-B).
- 2 cos A cos B = cos(A+B)+cos(A-B).
- 2 sin A sin B = cos(A-B)-cos(A+B).
- 2 sin A sin B is NOT sin(A+B)+sin(A-B). That one is 2 sin cos.
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