Compound Angles: Cos and Sin of Sums

Compound Angles: Cos and Sin of Sums

What is it?

Replace B by -B in the fundamental law: cos(A+B)=cos A cos B - sin A sin B (because cos is even, sin odd). Then sin(A+B)=sin A cos B + cos A sin B, from cos(pi/2 - (A+B)) or a similar deduction the book uses. sin(A-B)=sin A cos B - cos A sin B. Meaning: plus on the left for cosine becomes a minus between the products.

In simple words

• Replace B by -B in the fundamental law: cos(A+B)=cos A cos B - sin A sin B (because cos is even, sin odd).
• Then sin(A+B)=sin A cos B + cos A sin B, from cos(pi/2 - (A+B)) or a similar deduction the book uses.
• sin(A-B)=sin A cos B - cos A sin B.
• sin(A+B) is plus both. cos(A+B) is minus on the sin product.

Important terms

COMPOUND ANGLES: COS AND SIN OF SUMS: Replace B by -B in the fundamental law: cos(A+B)=cos A cos B - sin A sin B (because cos is even, sin odd). Then sin(A+B)=sin A cos B + cos A sin B, from cos(pi/2 - (A+B)) or a similar deduction the book uses. sin(A-B)=sin A cos B - cos A sin B. Meaning: plus on the left for cosine becomes a minus between the products.

1. COS(A+B): cos even: cos(-B)=cos B. sin odd: sin(-B)=-sin B. So the + in the difference law becomes a - between the sin product.

2. SIN(A±B): sin(A+B): both products plus. sin(A-B): minus on the second product. Memory: sine keeps the same ± as the left; cosine flips.

What you should learn

• State the main ideas and notation used in compound angles: cos and sin of sums.
• Apply the rules of compound angles: cos and sin of sums in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• cos(A+B)
• Check A=B: cos 2A = cos²-sin²
• Keep the minus

COS(A+B)

cos even: cos(-B)=cos B. sin odd: sin(-B)=-sin B. So the + in the difference law becomes a - between the sin product.

SIN(A±B)

sin(A+B): both products plus. sin(A-B): minus on the second product. Memory: sine keeps the same ± as the left; cosine flips.

Full school-level explanation

Replace B by -B in the fundamental law: cos(A+B)=cos A cos B - sin A sin B (because cos is even, sin odd). Then sin(A+B)=sin A cos B + cos A sin B, from cos(pi/2 - (A+B)) or a similar deduction the book uses. sin(A-B)=sin A cos B - cos A sin B. Meaning: plus on the left for cosine becomes a minus between the products.

Flip the sign of B, then sine:

WHAT — COMPOUND ANGLES: COS AND SIN OF SUMS

Replace B by -B in the fundamental law: cos(A+B)=cos A cos B - sin A sin B (because cos is even, sin odd). Then sin(A+B)=sin A cos B + cos A sin B, from cos(pi/2 - (A+B)) or a similar deduction the book uses. sin(A-B)=sin A cos B - cos A sin B. Meaning: plus on the left for cosine becomes a minus between the products.

RULE — 1. COS(A+B)

cos even: cos(-B)=cos B. sin odd: sin(-B)=-sin B. So the + in the difference law becomes a - between the sin product.

METHOD — 2. SIN(A±B)

sin(A+B): both products plus. sin(A-B): minus on the second product. Memory: sine keeps the same ± as the left; cosine flips.

WORKED EXAMPLE

Worked: sin 75 = sin(45+30). sin45 cos30 + cos45 sin30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(12) = sqrt(2)/2 * (sqrt(3)+1)/2 = (sqrt(6)+sqrt(2))/4

COS(A+B)

cos even: cos(-B)=cos B. sin odd: sin(-B)=-sin B. So the + in the difference law becomes a - between the sin product.

Remember

• even/odd

Examples

• cos(A+B)
• Check A=B: cos 2A = cos²-sin²
• Keep the minus

SIN(A±B)

sin(A+B): both products plus. sin(A-B): minus on the second product. Memory: sine keeps the same ± as the left; cosine flips.

Remember

• sine: same sign as left

Examples

• sin(A+B) both +
• sin(A-B) minus second
• Co-function if the book uses it

How to answer a compound angles: cos and sin of sums 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

• cos(A+B)=cos A cos B - sin A sin B
• sin(A+B)=sin A cos B + cos A sin B
• (sqrt(6)+sqrt(2))/4
• sine plus, cosine minus
• even/odd
• sine: same sign as left
• cos(A+B)=cos A cos B - sin A sin B
• sin(A+B)=sin A cos B + cos A sin B

How the ideas connect

Worked: sin 75 = sin(45+30). sin45 cos30 + cos45 sin30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(12) = sqrt(2)/2 * (sqrt(3)+1)/2 = (sqrt(6)+sqrt(2))/4

Deeper understanding

Go beyond memorising Compound Angles: Cos and Sin of Sums. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: sin 75 = sin(45+30). sin45 cos30 + cos45 sin30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(12) = sqrt(2)/2 * (sqrt(3)+1)/2 = (sqrt(6)+sqrt(2))/4

Common mistakes

Exam focus

Core Idea

1. COS(A+B)

2. SIN(A±B)

Worked Example

Exam focus

Quick revision

• cos(A+B)=cos A cos B - sin A sin B
• sin(A+B)=sin A cos B + cos A sin B
• (sqrt(6)+sqrt(2))/4
• sine plus, cosine minus
• COMPOUND ANGLES: COS AND SIN OF SUMS: Replace B by -B in the fundamental law: cos(A+B)=cos A cos B - sin A sin B (because cos is even, sin
• 1. COS(A+B): cos even: cos(-B)=cos B. sin odd: sin(-B)=-sin B. So the + in the difference law becomes a - between

sin(A+B) is plus both. cos(A+B) is minus on the sin product.

For long answers on Compound Angles: Cos and Sin of Sums: define clearly → explain the main parts → give one example → state one exam caution.