Product to Sum

Product to Sum

What is it?

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.

In simple words

• 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.

Important terms

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.

What you should learn

• 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.

Examples

• Prosthaphaeresis named
• 2 in front
• A=B recovers double

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.

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.

Full school-level explanation

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.

Sin-cos, then the two cosines:

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

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.

Remember

• expand compounds

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.

Remember

• sin sin: cos minus cos

Examples

• Both plus for cos cos
• Minus for sin sin
• Check A=B: 2 sin² A = 1-cos 2A

How to answer a product to sum 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

• 2 sin A cos B = sin(A+B)+sin(A-B)
• 2 sin A sin B = cos(A-B)-cos(A+B)
• 1 + sqrt(3)/2
• match the pair
• expand compounds
• sin sin: cos minus cos
• 2 sin A cos B = sin(A+B)+sin(A-B)
• 2 sin A sin B = cos(A-B)-cos(A+B)

How the ideas connect

Worked: 2 sin 75 cos 15. = sin 90 + sin 60 = 1 + sqrt(3)/2

Deeper understanding

Go beyond memorising Product to Sum. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: 2 sin 75 cos 15. = sin 90 + sin 60 = 1 + sqrt(3)/2

Common mistakes

Exam focus

Core Idea

1. 2 SIN COS

2. COS COS AND SIN SIN

Worked Example

Exam focus

Quick revision

• 2 sin A cos B = sin(A+B)+sin(A-B)
• 2 sin A sin B = cos(A-B)-cos(A+B)
• 1 + sqrt(3)/2
• match the pair
• 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
• 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

2 sin A sin B is NOT sin(A+B)+sin(A-B). That one is 2 sin cos.

For long answers on Product to Sum: define clearly → explain the main parts → give one example → state one exam caution.