MathematicsChapter 8: Fundamentals of Trigonometry
Sum To Product
The reverse: sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2).
Learning objectives
- State the main ideas and notation used in sum to product.
- Apply the rules of sum to product in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
The reverse: sin C + sin D = 2 sin(C+D2) cos(C-D2). sin C - sin D = 2 cos(C+D2) sin(C-D2). cos C + cos D = 2 cos(C+D2) cos(C-D2). cos C - cos D = -2 sin(C+D2) sin(C-D2). Meaning: a sum of two sines becomes a product. The cosine-difference identity has an extra minus — that is the one people drop.
Mathematical form
D = 2 sin(C+D2) cos(C-D2). sin C - sin
D = 2 cos(C+D2) sin(C-D2). cos C + cos
D = 2 cos(C+D2) cos(C-D2). cos C - cos
D = -2 sin(C+D2) sin(C-D2). Meaning
Where / symbols
- SUM TO PRODUCT
- The reverse: sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2). sin C - sin D = 2 cos((C+D)/2) sin((C-D)/2). cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2). cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2). Meaning: a sum of two sines becomes a product. The cosine-difference identity has an extra minus — that is the one people drop.
- 1. SIN PLUS/MINUS
- Let A=(C+D)/2, B=(C-D)/2, invert the product-to-sum. sin C - sin D: 2 cos(avg) sin(half diff). Order C then D matches the book's B = (C-D)/2.
- 2. COS PLUS/MINUS
- cos+cos: 2 cos cos. cos-cos: -2 sin sin. The minus is real: cos 0 - cos 60 = 1-0.5=0.5, and -2 sin30 sin(-30) wait use numbers carefully — check with C=60 D=0: cos60-cos0=0.5-1=-0.5, and -2 sin30 sin30 = -2*(1/2)*(1/2)=-0.5.
Overview
Sines, then cosines:
Sin Plus/minus
Let A=C+D2, B=C-D2, invert the product-to-sum. sin C - sin D: 2 cos(avg) sin(half diff). Order C then D matches the book's B = C-D2.
Examples sin+sin: 2 sin cos · sin-sin: 2 cos sin · Keep C minus D
Cos Plus/minus
cos+cos: 2 cos cos. cos-cos: -2 sin sin. The minus is real: cos 0 - cos 60 = 1-0.5=0.5, and -2 sin30 sin(-30) wait use numbers carefully — check with C=60 D=0: cos60-cos0=0.5-1=-0.5, and -2 sin30 sin30 = -2(12)(12)=-0.5.
Examples cos+cos both cos · cos-cos: -2 sin sin · Check with numbers
WHAT — SUM TO PRODUCT
The reverse: sin C + sin D = 2 sin(C+D2) cos(C-D2). sin C - sin D = 2 cos(C+D2) sin(C-D2). cos C + cos D = 2 cos(C+D2) cos(C-D2). cos C - cos D = -2 sin(C+D2) sin(C-D2). Meaning: a sum of two sines becomes a product. The cosine-difference identity has an extra minus — that is the one people drop.
RULE — 1. SIN PLUS/MINUS
Let A=C+D2, B=C-D2, invert the product-to-sum. sin C - sin D: 2 cos(avg) sin(half diff). Order C then D matches the book's B = C-D2.
METHOD — 2. COS PLUS/MINUS
cos+cos: 2 cos cos. cos-cos: -2 sin sin. The minus is real: cos 0 - cos 60 = 1-0.5=0.5, and -2 sin30 sin(-30) wait use numbers carefully — check with C=60 D=0: cos60-cos0=0.5-1=-0.5, and -2 sin30 sin30 = -2(12)(12)=-0.5.
WORKED EXAMPLE
Worked: sin 75 + sin 15. = 2 sin 45 cos 30 = 2(sqrt(2)/2)(sqrt(3)/2) = sqrt(6)/2
How to answer a sum to product 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
- sin 75 + sin 15.
- = 2 sin 45 cos 30
- = 2*(sqrt(2)/2)*(sqrt(3)/2)
- = sqrt(6)/2
In simple words
cos C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
Key points
- average and half-difference
- cos C - cos D has a minus 2
- avg = C+D2
- minus on cos-cos
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Sum To Product
Sticky Notes · Quick Revision
Core Idea
- The reverse: sin C + sin D = 2 sin(C+D2) cos(C-D2).
- sin C - sin D = 2 cos(C+D2) sin(C-D2).
- cos C + cos D = 2 cos(C+D2) cos(C-D2).
1. SIN PLUS/MINUS
- Let A=C+D2, B=C-D2, invert the product-to-sum.
- sin C - sin D: 2 cos(avg) sin(half diff).
- Order C then D matches the book's B = C-D2.
2. COS PLUS/MINUS
- cos+cos: 2 cos cos.
- cos-cos: -2 sin sin.
- The minus is real: cos 0 - cos 60 = 1-0.5=0.5, and -2 sin30 sin(-30) wait use numbers carefully — check with C=60 D=0: cos60-cos0=0.5-1=-0.5, and -2 sin30 sin30 = -2(12)(12)=-0.5.
Worked Example
- Worked: sin 75 + sin 15.
- = 2 sin 45 cos 30
- = 2*(sqrt(2)/2)*(sqrt(3)/2)
- = sqrt(6)/2
Common Misconception
- cos C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
FBISE Exam Check
- State the rule for sum to product before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- The reverse: sin C + sin D = 2 sin(C+D2) cos(C-D2).
- sin C - sin D = 2 cos(C+D2) sin(C-D2).
- cos C + cos D = 2 cos(C+D2) cos(C-D2).
- cos C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
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