MathematicsChapter 8: Fundamentals of Trigonometry
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).
Learning objectives
- 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.
Definition
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.
Formula
sin30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(12
Where / symbols
- 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.
Relationship
- cos(A+B)=cos A cos B - sin A sin B
- sin(A+B)=sin A cos B + cos A sin B
Overview
Flip the sign of B, then sine:
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.
Examples cos(A+B) · Check A=B: cos 2A = cos^2-sin^2 · 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.
Examples sin(A+B) both + · sin(A-B) minus second · Co-function if the book uses it
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
How to answer a compound angles: cos and sin of sums 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(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
In simple words
sin(A+B) is plus both. cos(A+B) is minus on the sin product.
Key points
- cos(A+B)=cos A cos B - sin A sin B
- sin(A+B)=sin A cos B + cos A sin B
- even/odd
- sine: same sign as left
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Compound Angles: Cos and Sin of Sums
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
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
Common Misconception
- sin(A+B) is plus both. cos(A+B) is minus on the sin product.
FBISE Exam Check
- State the rule for compound angles: cos and sin of sums before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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