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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.
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Sum to Product
What is it?
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.
In simple words
- 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.
Important terms
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.
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.
What you should learn
- 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.
Examples
- sin+sin: 2 sin cos
- sin-sin: 2 cos sin
- Keep C minus D
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.
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.
Recommended for Class 11
Full school-level explanation
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.
Sines, then cosines:
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
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.
Remember
- avg = 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.
Remember
- minus on cos-cos
Examples
- cos+cos both cos
- cos-cos: -2 sin sin
- Check with numbers
How to answer a sum to product question
How it works
- 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.
Remember
- average and half-difference
- cos C - cos D has a minus 2
- sqrt(6)/2
- minus on cos minus cos
- avg = C+D2
- minus on cos-cos
- average and half-difference
- cos C - cos D has a minus 2
How the ideas connect
Worked: sin 75 + sin 15. = 2 sin 45 cos 30 = 2(sqrt(2)/2)(sqrt(3)/2) = sqrt(6)/2
Deeper understanding
Go beyond memorising Sum to Product. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: sin 75 + sin 15. = 2 sin 45 cos 30 = 2(sqrt(2)/2)(sqrt(3)/2) = sqrt(6)/2
Common mistakes
Exam focus
Core Idea
1. SIN PLUS/MINUS
2. COS PLUS/MINUS
Worked Example
Exam focus
Quick revision
- average and half-difference
- cos C - cos D has a minus 2
- sqrt(6)/2
- minus on cos minus cos
- 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
- 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
cos C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
For long answers on Sum to Product: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- average and half-difference
- cos C - cos D has a minus 2
- sqrt(6)/2
- minus on cos minus cos
Important formulas
- average and half-difference
- cos C - cos D has a minus 2
- sqrt(6)/2
Common exam traps
- cos C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
- 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 C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
- 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 C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
average and half-difference
cos C - cos D has a minus 2
sqrt(6)/2
minus on cos minus cos
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
avg = C+D2
- 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
- Formula
- avg = C+D2
- Example
- sin+sin: 2 sin cos
- Critical warning
- cos C - cos D = -2 sin(C+D2) sin(C-D2). Do not drop the minus.
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Exam Practice
Define
Q1. Define Sum to Product.
Explain
Q2. Explain: SIN PLUS/MINUS.
Example
Q3. Give one clear example of Sum to Product and say why it fits.
Mistake
Q4. State one common mistake students make about Sum to Product.
Exam note
Q5. Write a short exam note on Sum to Product: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.